Translations of
MATHEMATICAL
MONOGRAPHS
Volume 136
Algebraic Geometry
Masayoshi Miyanishi
Translated by
Masayoshi Miyanishi
^-^,,
io American Mathematical Society
Providence, Rhode Island

DAISU KIKAGAKU
(Algebraic Geometry)
by Masayoshi Miyanishi
Copyright © 1990 by Shokabo Publishing Co.. Ltd.
Originally published in Japanese by Shokabo Publishing Co.. Ltd.. Tokyo in 1990.
Translated from the Japanese by Masayoshi Miyanishi
1991 Mathematics Subject Classification. Primary 14-01:
^Secondary 13-01. 14A10. 14A15. 14J99.
Abstract. This book covers algebraic geometry from the beginnings to an introduction of algebraic
surfaces, viz.. to the gate from which the classification of algebraic surfaces starts. The book has three
parts. The first part provides the necessary basic results from commutative algebras and the theory of
sheaves and its cohomologies. The second part is on schemes and algebraic varieties. The third part is on
algebraic curves and surfaces, placing emphasis on the use of linear systems and the associated rational
mappings.
Library of Congress Cataloging-in-Publicaiion Data
Miyanishi. Masayoshi. 1940-
[Daisu kikagaku. English]
Algebraic geometry/Masayoshi Miyanishi: translated by Masayoshi Miyanishi.
p. cm. — (Translations of mathematical monographs. ISSN 0065-9282: v. 136)
Includes bibliographical references and index.
rSBN 0-8218-4615-9 (acid-free)
1. Geometry. Algebraic. I. Title. II, Series.
QA564.M5713 1994 94-2018
516.3'5—dc20 CIP
© Copyright 1994 by the American Mathematical Society, All rights reserved.
Translation authorized by the Shokabo Publishing Co.. Ltd.
The American Mathematical Society retains all rights
except those granted to the United States Government.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
*.£ Printed on recycled paper.
Information on Copying and Reprinting can be found in the back of this volume.
This volume was typeset using AmS-T^X..
the American Mathematical Society's TgX macro system.
10 9 8 7 6 5 4 3 2 1 99 98 97 96 95 94

Contents
Preface to the English Edition ix
Preface xi
Part I. Preliminaries 1
Chapter 1. Theorem of Luroth 3
Chapter 2. Theory of Sheaves and Cohomologies 25
Part II. Schemes and Algebraic Varieties 59
Chapter 3. Affine Schemes and Algebraic Varieties 61
Chapter 4. Schemes and Algebraic Varieties 81
Chapter 5. Projective Schemes and Projective Algebraic Varieties 105
Chapter 6. Nonsingular Algebraic Varieties 133
Part III. Algebraic Surfaces 157
Chapter 7. Algebraic Curves 159
Chapter 8. Intersection Theory on Algebraic Surfaces 165
Chapter 9. Pencils of Curves 175
Chapter 10. The Riemann-Roch Theorem for Algebraic Surfaces 191
Chapter 11. Minimal Algebraic Surfaces 201
Chapter 12. Ruled Surfaces and Rational Surfaces 211
Solutions to Problems 221
Part I 221
Part II 226
Part III 230
List of Notation 237
Bibliography 241
Index 243

Preface to the English Edition
The interesting theory, where one first experiences the beauty in learning algebraic
geometry, is perhaps the theory of algebraic curves and surfaces. The theory of
algebraic surfaces is particularly important when one further studies higher-dimensional
algebraic varieties.
In order to approach this theory through modern algebraic methods, one is
required to have a basic knowledge of local rings, sheaves, cohomologies. projective
varieties, linear systems.of divisors, etc. This basic knowledge may include all results
given in introductory textbooks devoted to these subjects, and it takes some time to
get familiar with them.
This book grew out from the author's attempts to find a shortcut and thus, lead
students effectively to an understanding of algebraic surfaces. This book, originally
written in Japanese, is designed so that readers with only knowledge of algebra at
an undergraduate level can start learning algebraic geometry by themselves. The
description is. therefore, self-contained as much as possible.
The book consists of three parts. The first part gives background in commutative
algebras, sheaf theory, and related cohomology theory. The second part introduces
schemes and algebraic varieties, where one can learn the basic language of algebraic
geometry. The last part is devoted to guiding the readers to the gate from which the
classification of algebraic surfaces starts. It is most desirable that readers continue
to further their learning through other books on algebraic surfaces.
While preparing the English translation, the author is indebted to Drs. Koji
Yokogawa and Hiroyuki Ito for technical assistance in typesetting the manuscript
using LaTeX.
M. Miyanishi
June 1993

Preface
Students often say they get interested in algebraic geometry and start learning it
by themselves, only to find that there are too many prerequisites; they get lost in
preliminary results. The knowledge of algebra and geometry that is taught in college is
not sufficient for reading through most of the serious textbooks in algebraic geometry.
One needs knowledge of ring theory, field theory, local rings, and transcendental field
extensions, even sheaf theory and cohomologies with coefficients in sheaves, which
are certainly beyond the_level of college mathematics. But if one waits to begin the
study of algebraic geometry until after absorbing all the necessary knowledge, then
one may have lost interest.
So it is most desirable if such basic results appear in a textbook in algebraic
geometry when they are needed. In writing this book, the author tried to prove or
recall results when needed.
This book, meant for senior undergraduate students and for graduate students,
will guide students from the beginnings of algebraic geometry to an introduction of
algebraic surfaces, viz.. to the gate from which the classification of algebraic surfaces
starts. The readers are advised to read higher textbooks or research papers after
completing this book.
In order to keep the book to a reasonable size, some results were not included,
and even the included results are not stated in full generality.
The book consists of three parts. The first part provides the necessary basic
results. The second part is on schemes and algebraic varieties. The third part is on
algebraic surfaces. When we refer to results stated in the same part, we refer to them
in a form such as Theorem 3.1, and when we refer to results in different parts, we
include the part as in Theorem II.3.1.
At the end of each chapter; some results are given as problems; brief solutions
are given at the end of the book.
The author was encouraged to write this book by Professor S. Murakami of
Osaka University and given frequent and warm encouragement from Mr. S. Hosoki
of Shokabo Publ. Co. The author would like to express his sincere gratitude to both
of them.
M. Miyanishi
June 1990

Part I
Preliminaries

CHAPTER 1
Theorem of Liiroth
Let k be a field. If an extension field L of k is generated by finitely many elements
X] xn over k, it is called a finitely generated extension of k and is denoted by L =
k(x\ x„). For variables Zi X„. k[X\ Xn] denotes a polynomial ring in n
variables over k and k{X\ X„) denotes its quotient field. We call k(X\ X„)
a purely transcendental extension in n variables over k or an n-dimensional rational
function field, and call its elements rational functions.
Definition 1.1. Let L/k be a field extension. An element £, of L is called
transcendental over k or algebraically independent over k if £ is not algebraic over k.
For elements 6 6- of L. we say that 6 6- are transcendental or algebraically
independent over /c if £, is transcendental over A: (6 6-1) for each i. 1 < i < r.
The following result shows that the definition of 6 6- being transcendental
over k does not depend on the ordering of £,\ 6--
Lemma 1.2. For elements 6 6- o/L. the following conditions are equivalent:
(1) 6 6- are transcendental over k.
(2) For any permutation a of {1 r} and for each i. 1 < / < /*. £„(,) /.v
transcendental over £(67(1) 67(/-1))-
Proof. It suffices to show that (1) implies (2). Since any permutation is expressed
as a product of transpositions (i — 1. i), we have only to show the following:
For each /'. 1 < j < i, if ^ is transcendental over k{£,\ £/-i).
then 6_i is transcendental over fc(6 6-2-6)-
Suppose 6_i is algebraic over /c(6 £,-2- 6)- Then there exists an algebraic
relation
00(6-1)^+^1(6-1)^+--- + ^=0.
(*)
a,-G k[6 6-2-6]- a0aN^0. gcd(a0 aN) = \.
where a, ¢ fc[<Ji £,-_2] for some y. Let N/ be the ^,-degree of ctj. and write
°/ = ajo£i ' + (terms of lower degree in £,,). aj0 G £[6 6-2]-
Let M = max{yV;;0 < j < N}. Then we can write the relation (*) as
(**) J2 "/0(6-1 )N~j I 6M + (terms of lower degree in 6) = 0.

-I
1 I'RI-.LIMINARILS
Since £, .1 is transcendental over k{C\ £1-2)- we have
,V, A/
«/o(sB/-i)A-' ^0.
Hence, the relation (**) shows that £,- is algebraic over Ar(£) £; ,). This
contradicts the assumption (1). D
Lemma 1.3. Let L = k{x\ xn) be a finitely generated extension of k. and let
£1 £, he elements of L. Then the following assertions hold:
(1) If £ £,. are transcendental over k. then r < n.
[If no subsets of L strictly containing {c\ £, } are purely transcendental over k. then
{£1 £,-} is called a maximal system of transcendental elements and r is called its
length.)
(2) The length r of a maximal svstem of transcendental elements is a constant
independent of the choice.
(3) One can choose a maximal system of transcendental elements {n\ r\, } us
a subset of {x\ v„}.
Proof. As a subset of {x\ v,,}. we choose a system {t]\ r^} consisting
of elements transcendental over k by the following process applied to .v, x„ one
by one. If .V| is transcendental over k. we pick it up. If ai is algebraic over k. we
throw it away. Similarly, if x, is transcendental over k{x\ v,_i). we pick it up.
Otherwise, we throw it away. Applying this process from \\ to av we denote the
elements that we picked up by r\\ ns. Then .v < n. and Ljk{\][ 7J is an
algebraic extension.
Suppose now £1 £,. are transcendental over k. Since £] is algebraic over
k[rj[ nJ. we have
(*) tfn^O^+ai^i)^-1 +---+a,v -0. a, G k[r/\ n,\ uuaNf-0.
Since £1 is transcendental over k. a, <^k for some j. After relabelling {n\ n,}
if necessary, we may (and shall) assume that
ai = «/o>7i ' + (terms of lower degree in /71).
«/0.... £ k[tp_ rjx].
ala ^0 if a, ^0: Nj = 0 if a,- =0.
Set M = maxl/V^O < / < A'}, and rewrite (*) as
\
E «/o(Ci),V"'
N,-M J
n'{ — (terms of lower degree in n\
Then the coefficient of n\{ £ 0. Hence. t\\ is algebraic over k{c,].n2 nf). So £2 is
algebraic over k{£,\. r\i nf). Since £1 is transcendental over A:(£i )• the above
argument enables us to replace £2 by one of n2 n.,. say n2- One may thus assume that n2
is algebraic over k(£\. £2. rp ns). If L is algebraic over k{£\ C-ln \ 17,)
[i < /■). then one can repeat the above argument to £,-+1 because £,-_ 1 is transcendental
over /c(.£i £,) and replace £, m by one of 77,- + 1 ns. Hence, by induction, we
have r < s < 11.

1. THEOREM OF Ll'lROTH
5
Suppose {¢1 £,} is a maximal system of transcendental elements. Then L is
algebraic over k (¢1 £,,). If one exchanges the roles of {£,\ ¢,-} and {n\ ns]
in the above argument, one knons s < r. D
Definition 1.4. Let L = k{x\ v„) be a finitely generated extension of k,
A maximal system of transcendental elements {£\ ¢,} such as in Lemma 1.3 is
called a transcendence basis of L over k. and its length r is called the transcendence
degree of L/k. We write r = tr. deg/t L. If /• = 0. then Ljk is a finite algebraic
extension.
The following is a significant result which we often refer to in this book. The
proof is taken from Nagata [7].
Case (1) M/k is an algebraic extension. Let {c,\ ¢,-} be a transcendence
basis of Ljk. Since M{£,\ £,j-\)/k{£\ £j-\) is an algebraic extension. ¢, is
transcendental over M{£,\ £1-1)- Hence. ¢1 £,,- are transcendental over M.
Let {m\ m,\ be a set of elements of M which are linearly independent over k.
We shall show that m\ m, are linearly independent over k{£,\ £,,) as well.
Suppose
/i'"i +•••+./>, = 0. fj ek{c\ ¢,-).
Replacing /, /, by gf\ gf,. {g e /c[£i £,-])■ we may assume that /,
/, G^Ki £,]■ Writing
f, = ]T *<'>£"- C = C • ■ • C - « - (a, a,.). a}p e k.
U
we have Y^,„ Yl',-\ a„ m^" = 0. Since Yl'j--\ Un mi e ^ ail(^ £' £<■ are tran-
scendental over M. we have ^ — 0 for each a. Hence. a„ = 0 for each
a and each j. This entails /, = 0 for each j. So we have
t <[L:k(£_x £,.)]<+00.
The extension M/k is. therefore, a finite algebraic extension.
Case (2) M/k is not necessarily an algebraic extension. Choose a
transcendence basis {£| £,-} of L/k in such a way that {¢1 £a} is a maximal
system of transcendental elements of M/k. Then L/k{£\ ¢,,) is a finitely
generated extension, and M/k(£] £u) is an algebraic extension. Case (1) implies
[M : k{£_\ £,,)] < +00. Hence. M/k is a finitely generated extension. Since
{£\ ¢,,} is a transcendence basis of M/k. we have tr. deg/c M < tr. degA. L. □
Definition 1.6. Let L/k be a finitely generated extension. We say that L/k is
a separable extension if there exists a transcendence basis {£,] ¢,} of L/k such
that L/k{£,\ £,,.) is a separable algebraic extension. We then say that {¢1 £, .}
is a separating transcendence basis.
If the field k has characteristic zero, a finitely generated extension L/k is always
a separable extension. This is. however, not the case if the characteristic p of k is
nonzero. As an example, let/? = 2. and L = k{x.y). where x2 + y2 = a with a e k.

6
I. PRELIMINARIES
a1/2 ¢: k. Then L/k is not separable. In fact, (a- + y) <g 1 — 1 ® t/'/2 is a nilpotent
element in L ®/( Ar(a'/2). and hence. L/£ is not separable by the next result.
Lemma 1.7. Let L/k be a separable extension. Then we have the following
assertions:
(1) We can express L = k(£\ £,-.n) with F{£\ Zr-n) = 0- where r =
tr. deg;. L and F(X\ Xr. Y) is an irreducible polynomial over k.
(2) Suppose k has characteristic p ^ 0. Define a subfield kp of an algebraic
closure k of k by
kp" = {ap";a G k) (.v > 0).
Then there are no nilpotent elements in L (¾ kp for each s. s > 0.
Proof. (1) Let {£\ £,} be a separating transcendence basis of L/k. Then
L/k{£\ £,) is a simple extension since it is a separable algebraic extension. Hence.
we may write L = k{£,\ £,,--1]). where n is a root of a separable polynomial over
k{£x £,-).
G(£.Y) = a0YN +aiYN'1 +■■■ + aN = 0. a, e k{£, £,-)■
(2) Set k' = kp '. Since ci £,■ are transcendental over k'. k'®k k{£,\
c,) is a field, which we denote by k'{£,\ £,.). Then k'{£,\ £,r)/k{£,\ £,) is
a purely inseparable extension. Since
I«ii' = i%, =r)*'(£i 6).
we can consider L/k{£,\ £,) and k'{£\ £,)/k{£\ £r) instead of L/k and
k'Ik. respectively, and we have only to show that if L/k is a separable algebraic
extension and k'/k is a purely inseparable extension. L (¾ k' has no nilpotent
elements. If we write L = k[X]/(f{X)). then L®kk' = k'[X]/{f{X)). where f\X) is a
separable, irreducible polynomial over k. Hence. gcd(/(Z)./'(A')) = 1. This holds
also over k'. Hence. L (¾ k' is a field. So L <% k' has no nilpotent elements. □
Let L/k be a finitely generated extension. If condition (2) of Lemma 1.7 holds:
namely, if
(*) L (¾ kp has no nilpotent elements for each s > 0. then L/k is a separable
extension. For a proof, see Nagata [7]. Let k be an algebraic closure of k. We
say that L/k is a regular extension if L ®k k is an integral domain. Since k is a
flat ^-module (see Problem 1.1.9). the functor ®/(£ is injective. This implies that an
intermediate field extension M/k of a regular extension L/k is a regular extension
as well. If k is algebraically closed, then every finitely generated extension L/k is a
regular extension. By the above criterion (*) of separability, any regular extension
is a separable extension.
Lemma 1.8. Let L/k be a finitely generated extension. Then the following
conditions are equivalent:
(1) L/k is a regular extension.
(2) L/k is a separable extension, and k is algebraically closed in L.

1. THEORhM OF LI I ROTH
7
Proof. {1) implies (2). We shall show that k is algebraically closed in L. Suppose
an element a of L not in k is algebraic over k. Let f(X) be a minimal polynomial
of a over k. Then deg/(A") > 0. Write
f(X)=g(X>''). g'(X)^0.
Ill
g(X)=Y[(X-a,). a,ek.
i-i
Then /c(c/) <8>a- A is a subring of L (¾ A:, and
*(«) ®, A - k[X]/(f(X)) = £[*]/ f 11(^ - a,) J = flA[X]/(A-''' - a,).
Hence. k(a) (¾ A is not an integral domain, and this contradicts the hypothesis that
L (¾ k is an integral domain.
(2) implies (1). It suffices to show that L -¾ k' is an integral domain for a finite
normal extension k'/k. Set k" = {a <E k': a is purely inseparable over k}. Then
k" is a subfield of k'. and k'/k" is a Galois extension. Since L/k is a separable
extension. L cgik k" is a field and L (g>k k"/k" is a separable extension. We shall
see that k" is algebraically closed in L (¾ k". Suppose a" = J^; b; (g> a;, {bA € L.
a, e k") is algebraic over k". Since a'- G k (for each A) for some integer ,v > 0.
a"p' = Y^?M a1 ® 1 's an element of L and algebraic over k. Hence. «"''' G /c by
the hypothesis. If a" ^ A", then L <g.k k" would not be separable over k". Hence.
a" e k".
It suffices, therefore, to show that
If k'/k is a Galois extension, then L -¾ k' is a field.
Let m be a maximal ideal of L ®k k'. The residue field L (g:k k'/m then contains
L and k' as subfields and is generated by L and k'. We write L (¾ A'/m as L(k') and
call it a composite field of L and k'. We claim that L(k')/L is a Galois extension
and [L(A') : L] = \k' : A]. In fact, write k' = k(a). and let a\ — a. ai a, be all
conjugates of a over k. Then L(k') ~ L(a) and L(k')/L is a Galois extension because
a\ are L(k'). Let /(A') = [}/-_ i (A'-",), and let/i (A') be a minimal polynomial
of « over L. Then /i(A")|/(A"). Since we can write f\(X) = n)-i(^ _ "',)• ^e
coefficients of /i (A*) are polynomials in <;/„ a,- . Hence, they are the elements of
L n k'. Since L n A' = k by the hypothesis, we have f\ {X) G k[X]. This implies
/l(A') = f(X). So [L(A') : L] = [k': k]. and the claim is verified. Since the natural
surjection n: L <g>k k' —> L(k') is an L-homomorphism and [L ®/; k' : L] = [A' : /c].
the homomorphism 7r is an isomorphism. L ®K k' is. therefore, a field. □
If one replaces the field extension L/K by a ^-algebra A in the foregoing, what
kind of results can one expect? We shall consider this problem. In the following, a
ring signifies a commutative ring with identity element 1.
Definition 1.9. Let S be an integral domain and let R be a subring. If an
element x of S is a root of a monic polynomial f{X) = X" + a\X"~[ + ■ ■ • + a„
with coefficients in R (i.e.. / (a) = 0). we say that .v is integral over R. If every
element of S is integral over R. we say that S is integral over R. or S is an integral
extension of R.

8
]. PRELIM1NARIHS
Lkmma 1.10. Let S be an integral domain, and let R be a subring. Then the
following assertions hold:
(1) The following three conditions on an element x of S are equivalent:
(i) x is integral over R.
(ii) R[x] is a finitely generated R-module.
(iii) There exists an R-subalgebra R' of S containing x whieh is a finitely
generated R-module.
(2) If elements x. y of S are integral over R. then x ±y and xy are integral over
R. Hence. R = {x € S: x is integral over R} is an R-subalgebra of S.
(3) If S is integral over R. then the following two conditions are equivalent:
(i) R is a field.
(ii) S is a field.
(4) For integral domains T D S D R. if S is integral over R and T is integral
over S then so is T over R.
Proof. (1) (i) implies (ii). Suppose f{x) = x" + alx"'1 -\ \- a„ = 0 (a,■ e R).
Then R[x] = R ■ I + Rx + ■ ■ ■ + Rx"~]'.
(ii) implies (iii). Take R[x] as R'.
(iii) implies (i). Write R' = £^_t Rmh Then we can write xm, = Y^'j^-i a'imi
{ujj 6 R). The matrix representation of this relation is A'{m\ m,) = 0. where
A = (xdjj - an). Let A be the cofactor matrix of A. and let d = det/4. Then
' A A - dEt. where E, is the identity matrix. Hence, dm, = 0 for each j. On the
other hand, since one can express 1 = ^/ -i ^im> (^ £ R)- d = X!/-i bidm, ~ 0-
The expansion of d yields a monic relation of x.
d =x' -Tr{uu)x'-1 +---+det(-(tf,7)).
Hence, x is integral over R.
(2) Since both R[x] and R[y] are finitely generated ^-modules. R[x.y] is also
a finitely generated ^-module. We then know by (iii) of (1) that x ± y and xy are
integral over R.
(3) (i) implies (ii). It suffices to show that any nonzero element x of S has an
inverse a'"1. Since x is integral over R. we have
x" +u]x"~] +•••+«„ = 0. a; G R.
We may assume here that a„ ^ 0. Then
1 = ~a;] (.v"""' + aix""2 + • • • + «„-i)x with a,;1 e #:
whence, .v "' = -^'(.v"-1 + a\x"~2 H + c*„_i) € S.
(ii) implies (i). Let y be a nonzero element of R. Then _y_1 e S and _>'"' is
integral over R. Hence.
(r'T + ^t/'r-' + .-. + A^o. b,GR.
We then have 1 = ->'(ft, +• • ■ + ft,„v'"-1): whence. v_1 = -(/>, + • • • +Vv'"'-1) £ ^.
This shows that R is a field.
(4) Let z be an element of T. Since r is integral over S. we have z" + c\z"~x +
• • • + <"„ = 0 (c, e S). Hence. R[c\ c„.z] is a finitely generated R[c\ <•„]-
module. Meanwhile. R[c\ e„] is a finitely generated ^-module because c\ c„

1. THI.ORFM OF LI!ROTH
9
are integral over R. By (iii) of (1) above, we can conclude that z is integral over
R. □
Definition 1.11. The subring R of S defined in (2) of Lemma 1.10 is called the
integral closure of R in S. If R = R. we say that R is integrally closed in S. When S
is the quotient field Q(R) of R. R is called the normalization of R. If R is integraly
closed in Q{R). we call R an integrally closed integral domain,
We shall next recall basic results on noetherian rings.
Lkmma 1.12. The following four conditions on a ring are equivalent:
(i) For an arbitrary ascending chain of ideals Iq C I\ C ■ • ■ C /, C • • •. there exists
N > 0 such that In = /#+i = • • •. (When this condition is satisfied, we say that R
satisfies the ascending chain condition.)
(ii) Introduce in the set {1:1 is a proper ideal of R} an order I < J by inclusion
/C/, Then an arbitrary subset 6 has a maximal element. {When this condition is
satisfied, we say that R satisfies the maximal condition.)
(iii) An arbitrary ideal of R is a finitely generated ideal.
(iv) An R-submodule of a finitely generated R-moduk is a finitely generated R-
module.
Proof, (i) implies (ii). We have only to prove that 6 is an inductively ordered
set. Then we can apply Zorn's lemma to 6. Meanwhile, if {/, }/Ga is a totally ordered
subset of 6. the ascending chain condition obviously implies that it has the greatest
element.
(ii) implies if). Let /o C /) C • ■ • C I,■ C • • ■ be an ascending chain of ideals.
Then the set & = {/,: i > 0} has a maximal element IN which must be the greatest
element.
(i) implies (iii). Let I be an ideal of R. If I ^ (0). then take «0 G / - (0) and set
Iq = («o). If / ^ /(). take a\ e I — /0 and set f = («().«i). Repeating this process,
we can find an ascending chain of ideals /o C /, C j2 C ... _ but this process must
terminate if R satisfies the ascending chain condition. Hence. I — («0- «i «#)■
(iii) implies (i). Let /o C I\ C • • • C I,■ c • • • be an ascending chain of ideals.
Let I = \Ji>0Ii- Then I is an ideal of R. By the condition (iii). I is written as
I — {a0.a\ (7,,). Then there exists N > 0 such that a, £ IN for each i. Then
In = In + \ =
(iii) implies (iv). Let M be a finitely generated ^-module, and let N be an R-
submodule. We write M = Rz\ H 1- R:,„. Consider the case m = 1. Let I = {a e
R: az\ e N}. Then I is an ideal of R. So by (iii). we can write I = (a\ a„). For
any v e N. write v = az\. Then a £ I and a = a\a\ + ■■■ + a„a„. Hence.
v = a\{a\Z\) + ■■■ + a,,(a„z\). So N C J^"_, /?(«,-n). Besides. a,:\ G TV for
each i. and hence. J^"_, ./?(a,ri) C N obviously. So N - J2" -\R(a:-\) and N
is a finitely generated ^-module. If m > 1. we shall proceed by induction on m. Set
M\ = Rz2 -\ 1- Rz,„ and N\ = Nf)M\. By the induction hypothesis. N\ is a finitely
generated fl-module. So write N\ = £/ = i Rhi- Set 7 = {« G ^: («-i +M\)f~)N f- 0}.
Then / is an ideal of R. If we write / = (a\ as) by the condition (iii). there
exist elements #i g,s of N such that gj — a-:z\ 6 Mi for each _/'. Take an arbitrary
element z of N and write r ~ a\Zi H 1- Q,„r,„. Then Qi G /. So if we write Q| -
/?,«,+■■ • +&«,. -" - E)-ift«/ e M, H7V = /V,. Hence. - e ^)_, %, + ^- , ^/-
Therefore, we have N = ^)-\ Rg, + E/-i ^-

0
I. PRELIMINARIES
(iv) implies (iii). R is a finitely generated /^-module, and any ideal I of R is an
.R-submodule. Hence, the condition (iv) implies apparently the condition (iii). D
By definition. a ring R is a noetherian ring if R satisfies the above four conditions.
If an /^-algebra S is generated (as an ^-algebra) by finitely many elements f\ /„.
we write S = R[f \ ./'„] and call S a finitely generated ^-algebra. S is then the
image of an ft-homomorphism from a polynomial ring R[x\ x„] over R to S
defined by a,- h^ /', for each i.
Lkmma 1.13. Let R he a noetherian ring. Then the following assertions hold:
(1) The residue ring R/I is a noetherian ring.
(2) Let S he a multiplieatively closed subset of R. Then the quotient ring Rs
[denoted also by S~xR) is a noetherian ring.
Proof. (1) The assertion follows from the fact that the correspondence of ideals
{J: an ideal of R. J D /} > {J: an ideal of R/I} given by J ^ J/I is a one-to-one
correspondence which preserves the inclusion relation.
(2) The correspondence of ideals {J: an ideal of R. J n S = 0} —> {J': a proper
ideal of /?.«,} given by Ji->JRs is a surjective mapping which preserves the inclusion
relation. The assertion follows from this fact. [.1
Lemma 1.14. A polynomial ring R[.\\ \„] over a noetherian ring R is also
noetherian. Hence, a finitely generated R-algehra is noetherian.
Proof. By induction on n. it suffices to prove the assertion in the case n — \.
Let / be a proper ideal of R[x]. and let
a = {c € R: for some /(.v) 6 I ■ fix) = ex" + (terms of lower degree)}.
Then a is an ideal of R. Write a = E/m a>R- and let ./'/ = a>x"' + (terms of lower
degree) be an element of/ corresponding to a,. Set m - max{;;,: 1 < / < r}.
Replacing f, by j -,x"'~":. we may assume n\ = • • • = n, = m. Let M = R - 1 + Rx +
■ ■ ■ [ #.\"'_l and A' =- I n M. Then M is a finitely generated ^-module, and we can
write N = Ylj_\ Rp,j■ We shall then show that I = £',■'.., R[x].f) + £',-1 #[*k/• For
g € I. g = ex" + (terms of lower degree), where c € a. Write e = ot\ci\ H f a,ar.
If « > m. deg(# - V'-_i a'//-v"' '") < « and £ - E/ = i otjfix"-'" fc /. Continuing
this argument, we find A] /7, e i?[.v] such that deg(# - J^ 1 ^///) < '"■ Then
g ~ T.'', , hj, € / r, M - Ar. So. g € C ., R[x]f, A £',_, *[*]*/■ Then we know
The following is an analogue of Lemma 1.3 in ring theory.
Theorem 1.15 (Noether's Normalization Theorem). Let k be afield, let R be
a finitely generated k-a/gebra domain1 and let L be the quotient field of R. Then there
exists a transcendence basis {\\ x„} such that x\ v„ G R and R is integral
over u polynomial ring k\x\ x,,]. where n = tr. degA L.
Proof . Since R is a finitely generated A-algebra domain, we can express R as the
residue ring R = k[Y\ Y,,,]/^ of a polynomial ring, where m > n and ^3 is a prime
ideal. Consider first the case m — n. Let y-, = 7, (mod^3). Then R = k[y\ y„]
and L = k(y\ v„). Since we can find a transcendence basis of L/k as a subset
A A-algebra domain 1110:111¾ a fr-algebra which is an integral domain.

1. THEOREM Ol' LLIROTH
II
of {y\ y„} and since n = tr. deg,. L. we know that y\ y„ are algebraically
independent over A:. If ^5 ^= (0). then any nonzero element F(Y\ Y„) of ty would
give a nontrivial algebraic relation F(y\ y„) = 0. This implies ty = (0). Namely.
R is a polynomial ring over k. Consider next the case m > n. Since ty ^ (0). there
exists a nontrivial algebraic relation f(y\ >•„,) - 0. Set r, = y-, — y[' (2 < i < m).
where we choose r,-'s in such a way that 0 « f] «/•.]<•■■ « r,„. Then we have
f{y\ yn,) = f [y\ ■ -i + y\2 -,» + y'{")
(terms of lower degree in vi
1 I with coefficients in k[z2 :„,]
where b e k* = k - (0). Hence. y\ is integral over k[z2 :„,]■ Since j>,- =
Z-, + y\' ■ y, is integral over k[:2 :m] as well for 2 < i < m. Namely. R is integral
over k[zi -,„]. Consider next k[z2 z,„] instead of R. Then by induction on
m - n. we know that there exists a transcendence basis xi x„ as required in the
assertion. □
In order to state results which are derived from Theorem 1.15. we need the
following definition and some preparations.
Definition 1.16. Let R be a ring. A prime ideal sequence of length n is an
ascending chain of prime ideals of R
Pogpi g ••■ gp„.
The supremum of the length of a prime ideal sequence is called the Krull dimension
of R and denoted by K-dim./? (which could be oo). When R is a finitely generated
algebra domain over a field k. the transcendence degree of the quotient field Q{R)/k
is called the dimension of R and is denoted by dim R. Whenever we know K- dim R =
dimR. we represent it by dim./?.
We shall summarize basic results on maximal ideals in the following:
Lemma 1.17. Let R be a ring. We then have the following assertions:
(1) Given a proper ideal I of R. there exists a maximal ideal containing I.
(2) For a proper ideal m. m is a maximal ideal if and only if R/m is afield. A
maximal ideal is. therefore, a prime ideal. The ring R has a unique maximal ideal m if
and only if all elements of R — m are invertible elements. {If R has a unique maximal
ideal m. R is called a local ring and is expressed by a pair (R.m).)
(3) Ifp is a prime ideal, then R — p is a multiplicatively closed set. and the quotient
ring Rp of R with respect to R - p is a local ring with maximal ideal pRp.
Proof. (1) Given an ideal /. let 6 = {J: I C J c r} be the set of proper ideals
containing I. Introduce an order > by the inclusion relation: J2 > J\ if and only if
Ji 2 J\■ Then with this order. 6 is an inductively ordered set. So by Zorns lemma,
there exists a maximal element m in 6. and m is a maximal ideal of R containing I.
(2) m is a maximal ideal if and only if (0) is the unique proper ideal of R/m.
This is equivalent to saying that R/m is a field. It then follows that R/m is an integral
domain. Hence, m is a prime ideal. If m is a unique maximal ideal, then we have
xR — R for each x € R - m; whence, x is invertible. The converse follows immediately
from the next observation:
= 0.

I. PRELIMINARIES
If there exists a maximal ideal m' other than m. any element of m' — m
is not invertible.
(3) follows from (2). □
Lkmma 1,18. Let S be an integral domain, and let R he a subring. Assume that
S is integral over R. Then we have:
(1) (Lying-over Theorem). For an arbitrary prime ideal p of R there exists a
prime ideal 2J of S such thattynR = p. //'23' is a prime ideal of S with 03 C 03' and
03'ntf = p. then 03 = 03'.
(2) (Going-up Theorem). For any prime ideal sequence
Po § Pi g • • ■ g P»
of R there exists a prime ideal sequence
% % Vi § • ■ ■ s %,
of S such that 03,- n R = p,- (0 < i < n).
(3) We have the equality K-dimS = K-dimi?.
Proof. (1) Let T = R - p. Then T is a multiplicatfvely closed subset of R
and S. Suppose 03 is a prime ideal of ST such that 03 n Rp = pRv. Then we have
(($nS)r\R = <$nRf,riR = pRp(lR = p. and 03 05" is a prime ideal of 5 as required.
So by replacing R and 5 by Rp and Sr. respectively, we may (and shall) assume that
R is a local ring with maximal ideal p. Let m be an arbitrary maximal ideal of S.
and let q = m n R. Then R/c\ is a subring of 5/m and S/m is integral over R/q.
So R/q is a field by Lemma 1.10 (3). Namely, q is a maximal ideal. Hence, q = p.
Then we take m as a prime ideal 03 which we require. Next suppose 03 n R ~ 03' n R.
Replacing as above R and S by Rp and S, . respectively, we may assume that (f.p)
is a local ring. Since S/23 is an integral extension of a field R/p. 03 is a maximal
ideal of S. So 03 = 03' provided 03 C 03'.
(2) By induction on n. we have only to treat the case n = 1. By (1) above, there
exists a prime ideal 2¾ of S such that 2J0 n R ~ po. By replacing S and R by ST and
RVl respectively, we may assume that (R. pi) is a local ring, where T = R - p\. Here
2Jo is not a maximal ideal of S because 5/030 is an integral extension of R/po and
R/po is not a field. As 23|. take a maximal ideal of S containing 230. The argument
in (1) above then shows that 031 n R = p i.
(3) Given a prime ideal sequence 03o ^ 231 ^ • ■ • ^ 23,, of S. the sequence
po ^ pi 9 " ' § P" w^ P' = 23, H ft is a prime ideal sequence of R. Hence.
K-dim.fi < K-dimS. The converse inequality follows from (2) above. □
After these preparations, we shall prove the next result.
Theorem 1.19. With the same assumptions as in Theorem 1.15. we have K-dimft =
dim/?.
Proof. We shall prove the assertion by induction on n. If n — 0. then R is
a field and the equality holds obviously. Assume that the equality holds provided
dim ft < n - 1. Consider first the case where R is a polynomial ring k[x\ v,,].
Let (0) = po C p, C ... C pr be a prime ideal sequence of R. Then R = R/p\ is
a finitely generated /r-algebra domain, and Q(R) ~ k(x\ x„). where .v, = x,
(modpi). Since a nonzero element in pi provides a nontrivial algebraic relation
among x\ v„. we have tr. degAQ(R) < n. By the induction hypothesis, r —

I. THF.ORFM OF LUROTH
13
1 < n - 1. Hence, r < n. This implies K-dim ft < dim ft. Besides, we have a
prime ideal sequence (0) C (x\) C (.vi..v2) J '" § (■*' v») °^ ^: whence.
K- dim ft > «. Therefore. K- dim R — dim R = n. If R is not necessarily a polynomial
ring. R contains a polynomial ring k[x\ a„] and R is an integral extension of
k[x\ x„\. By Lemma 1.18 (3). we know that K-dim R = \L-<Ximk[x\ a„] =
n. □
We shall next recall a decomposition of an ideal into an intersection of primary
ideals in a noetherian ring.
Definition 1.20. A proper ideal q of a ring R is a primary ideal if it satisfies the
condition
ab G q. a ¢. q implies h" £ q for some n > 0.
Taking the contrapositive. it is equivalent to saying that
ab 6 q. b" £ q for every n > 0 implies « 6 q.
An ideal / of ft is a reducible ideal if / = Ji n Ji with two proper ideals J\. J2Q /)•
An ideal which is not reducible is called an irreducible ideal.
Lemma 1.21. (1) If q is a primary ideal the radical ideal y/q = {a e ft: a" 6 q
/r;r .s't»we « > 0} w a prime ideal. (1^? .raj> that ^/q is the prime ideal associated to q
and q belongs to ^/q.)
(2) Let q 6t> «« jfifcw/ of R. If q D m" (/w .swwe « > 0) wrY/i c/ maximal ideal m.
ffc« q is a primary ideal.
(3) Let q, (1 < / < r) be primary ideals belonging to one and the same prime ideal
p. Then qi n ■ ■ ■ fl q, is a primary ideal belonging to p.
Proof. (1) If ab e ^q and b ¢. yfq. then (ab)r £ q and (br)" ¢. q (for every
n > 0). So ar e q. i.e.. a 6 ^/q.
(2) Obviously, ^q = m. Suppose ab e q and /) ^ ^/q. Since />ft + m" = ft. we
have hx + y — I for x e ft and j> € m". Then a = abx + ay e q.
(3) If ab eqi n- ■ -nq,. and b ¢. p. then a e q, (for each i). Hence, a e q\ n- • -flq,-.
In general, we have v7i n ■ ■ ■ n /, = y/J\ n • • • D vX; whence, ,/qi n ■ ■ • n q,- = p. □
Lemma 1.22. Let R be a noetherian ring. Then we have:
(1) An arbitrary proper ideal I of R is written as the intersection I = J\ fl • • • fl J,
of finitely many irreducible ideals J\ J„.
(2) An irreducible ideal I is a primary ideal.
Proof. (1) We proceed by reductio absurdum. Let & be the set of proper ideals
of ft which cannot be written as the intersection of finitely many irreducible ideals.
Suppose & is not the empty set. Introduce an order in G by the inclusion; that
is. I\ < I2 if I\ C /2. Since ft is a noetherian ring. 6 has a maximal element by
Lemma 1.12. Let J be a maximal element of 6. Then J is a reducible ideal. Hence.
J = J\ n Ji for ideals J\. J2 {J ^ Jj ^ ft. / = 1.2). Since J, ¢. &. we may write
Jj — Jj\ fl • • • n Jj,-, as the intersection of irreducible ideals. Then J = f]/=n (T/-i ^7
which is the intersection of irreducible ideals. This contradicts the choice of J.
(2) Suppose abel and b" £ I (for each n > 0). Let 7„ = {.v e ft: b"x <E /} (which
we denote also by (I :b")). Then J„ is a proper ideal of ft and Ji QJ2C---cJ„C---.
Since ft is a noetherian ring. J,„ = Jm+] = ■ ■ ■ for some m > 0. Let /i = / + aR
and /,=/+ />'"ft. Then /C/,n /2. We shall show that /,n/2C /. Write an

I
14 I PRF-L1MINARILS
element _v of I\ n f as _v - 0 + ay =- (i f- /?"'- with t*. /? e / and v. : & R. Then
/>'" ''- = />(a -/*) + «/u- G /: whence, r G 7,iH 1 = J,„. So />'"; € / and .v G /.
accordingly. This implies / - /] n /;. Since / is an irreducible ideal by hypothesis
and since h =^ I. we have 1 = 1]. Hence, a G I ■ □
As a corollary of Lemma 1.22. we have the next result.
Thhorhm 1.23. (1) An arbitrary ideal I of a noetherian ring R has a primary ideal
decomposition I = qi D ■ • ■ n q„ satisfying the following two conditions:
(ii) / ^ q, n • • ■ 1 q,- H • • ■ n q,„ /<;/■ «h7j ;'. h'/ktc cj; means that q, ;.s omitted.
[We call such a decomposition the shortest expression by primary ideals.)
(2) Let I — qi H • • ■ H q„ be the shortest expression of I by primary ideals. Then
the set of ideals {v%: 1 < i < «} coincides with the set {p; p is a prime ideal of R.
p = {/ : a) for some a G R}. This set is. therefore, determined uniquely by the ideal I.
{Here (I : a) = {.v G R: ax <= /} by definition, and we call it an ideal quotient of I
by (a). We call v'q7 a prime divisor of I.)
(3) Ld'f {pi p, \ exhaust all minimal prime divisors ofl. Then \JI — pi n- ■ -np,-
«/«/ >/7 C p, n ■ ■ ■ n p; n - ■ - n p, for each i.
[This decomposition is called the prime divisor decomposition of \/1.)
Proof. (1) By Lemma 1,22. there exists a primary ideal decomposition I -=
qi H • ■ ■ n q„. If (\\ n ■ ■ ■ fi q, n • • ■ n q„ = I. we consider a decomposition of I with
qj. omitted. Hence, we may assume that no q, of I — q\ n • ■ • n q„ is redundant. For
any prime ideal p in {y/qT: 1 < / < «}. the intersection of all q,'s with y/qj = p is a
primary ideal by virtue of Lemma 1.21. So we may assume that the condition (i) is
satisfied as well.
(2) Suppose (I : a) equals to a prime ideal p. Then (/:«)= p)" , (q, : a).
(q,- :«)Cp (for some i). Since p c (q, : a) also, we have p = (q, : a). Note here that
if q is a primary ideal and (q : a) 7= R. then \/(q : a) = sjq. (In fact. xfq C \/[q : a)
because q C (q : a). For .v G \/{q : a). ax* G q for some N > 0. and .v"" G q
for some A/ > 0 because a <£ q. Hence, x G y'q.) Thereby we know that p = y/qj.
Conversely, suppose p = v/q7. We shall show that p = (/ : a) for some a g R. Take
an element a G qi P • • • n q, n • ■ • n q„ - I. Then (/:«) = (q,- : u). Moreover, since
((/ : a) : b) = (/ : ab) and (q,- : a) is a primary ideal belonging to p. we have only
to show that p — (q : a) for some a G R provided q is a primary ideal belonging
to p. Let 6 - {J: J = (q : a), q C J C p} and define in 6 an order by J, < J2
if V| C 72- Here 6 === 0 because c ^ p entails (q : c) = q G 6. Moreover, since R
is a noetherian ring. S has a maximal element. We shall show that if.(q : a) is a
maximal element, then p — (q : a). Suppose (q : a) C p. Then since p = \f[q\ a).
there exists an element b of p - (q : a) such that b ¢. q : a) and h- G (q : a). Then
(q ; a) ^ (q : w/>) C p. This contradicts the maximahty of (q : a). Hence, p = (q : a).
(3) Let I = qi n ■ ■ ■ n q„ be the shortest expression by primary ideals. Then
v'7 = y/qT n ■ ■ • n y/q^. If {pi p,,} is the set of minimal prime divisors of /. we
have apparently •/! = p, P ■ ■ ■ np,-. If pi n ■ • -Pp,- P • ■ • np, = \fl. then p, 2 rj/// Pi-
and hence, p, 2 />/ for some / 7^ i. which is a contradiction. D
A nonminimal prime divisor of / is called an embedded prime divisor of /. We
shall explain in the Part II what the prime divisor decomposition of \fl means
geometrically.

1. THKORFM OF Ll'lROI'H
IS
ThhORFM 1.24. (1) Let I be a proper ideal of R. Then \fl - (~}p. where p ranges
over all prime ideals containing I,
(2) (The Nullstellensutz of Hilbert). Let R be a finitely generated algebra over
a field k. and let J be a proper ideal of I. Then \// — f]m- where m ranges over all
maxima! ideals containing I.
Proof. (1) Apparently. \fl C p|PD/ P- Suppose .x cf)P «<nd .v ¢. \fl, Then
S = {a': i > 0} is a multiplicative!}' closed subset of R with 5 n / = 0. where .v" = I.
Since IRS j= Rs in the quotient ring Rs. there exists a maximal ideal OT of Rs
containing I Rs- Let p' = {a c R: a/\ e 971}. Then p' is a prime ideal of R such
that p' 2 / and p' n S — (d. However, this is a contradiction because .\ 6 pjp C p'
and .v c S. Hence f]„DI p = \/7-
(2) It suffices to show that if x is an element of R with .v ¢. \fl. then there
exists a maximal ideal m of R such that /Cm and x ¢. m. For such an element
.v. let 5 = {x'\ i > 0} as in (1) above. Then S is a multiplicatively closed subset
of R. and Rs is a finitely generated algebra over k. Since I n S = 0. we have
I Rs 7^ /?s. Let 9Jt be a maximal ideal of Rs containing I Rs and let m - {a £ R:
a/1 e 99?}. Then /Cm and .r ^ m. We shall show that m is a maximal ideal of
R. Let R = R/m. Then R is a finitely generated algebra domain over k and the
residue class y of x is a nonzero element of R. We shall show that dim R[l/y] = 0.
Suppose dim R[\/y] > 0. Then there exists a prime ideal ¢5 of /? with y $ ^5. Let
<P = {a C /?; a(modm) C *#}. Then <p is a prime ideal of R such that .v ¢ ^3 and
m C rp. Hence. OT — m.R.s ^ ^/?.v ?^ ^.s which contradicts the maximality of SOL
So dimK[l/v] = 0. and /?[l/y] is a field. Then we conclude by Lemma 1.25 below
that R is a field. Accordingly, m is a maximal ideal of R. LI
Lemma 1.25. Let R be a finitely generated algebra domain over a field k. and let
y be a nonzero element of R. If R[\/y] is a field, then so is R.
Proof. Let n -■- dim/?. We have only to show that n 0. Suppose n > 0. By
Noether's normalization theorem. R is an integral extension of A = k[x] \-„].
Since y is integral over A. we have
;■'" + a]y"'-[ +■■■ + a,„ = 0. «,- C A. am ^ 0.
Since /?[l/y] is a field by hypothesis, y G ^J for any prime ideal Cp(/ (0)) of/?. Recall
that for any prime ideal p(^ (0)) of A there exists a prime ideal <p of/? with tyoA = p.
Since y G %$. we have a„, 6 p. Note also that since A is a unique factorization domain
(UFD). any irreducible element f of A is a prime element. Namely, the principal
ideal (/ ) is a prime ideal of A. So by the foregoing argument. u„, E (/ ). Hence.
/ is a prime factor of a,„. Therefore. A has only finitely many irreducible elements.
This is a contradiction. (Prove that A has infinitely many, mutually prime, irreducible
polynomials.) Hence, n = 0. L2
In the subsequent paragraphs, we shall describe the relationships betwen normal
rings and discrete valuation rings.
Definition 1.26. (1) A noetherian integral domain R is called a normal ring if
it is integrally closed.
(2) Let & be a subring of a field L. & is called a valuation ring of L if & satisfies
the following two conditions:
(i) & + L.

16
I. PRI LIMINARIF.S
(ii) for each .v e L. either .v e L or .v"1 6 L.
A valuation ring 0 is called a discrete valuation ring (abbreviated as DVR) if it
is noetherian.
Lkmma 1.27. Let L he a field, and let & be a valuation ring of L.
(1) Set m = {.v G 0: x = 0 or .v"1 ¢0}. Then {<f.m) is a local ring.
(2) L /.v the quotient field of'&. and if is an integrally closed, integral domain.
(3) (f is a discrete valuation ring if and only ifm is principal, i.e.. m = ttf. and
n„>«m" = (0).
(4) For a discrete valuation ringed, define a mapping v: L^Zu (oo) by c(0) = oc
and v(x) = n if and only if xt~" e if - m for x e L* := L - (0). Then v
satisfies the following properties:
(i) v(x) = oc if and only if x = 0.
(ii) v(xy) = v{x) +v{y).
(iii) v(x+y) >mm(t,{x).v{y)).
In terms of the mapping v. one can write & = {.y G L: t'(.v) > 0} and
m = {.v G L: 7»(.v) > 0}. [v is called the discrete valuation associated
with if.)
Proof. (1) We shall show that v. y g m implies x + y G m. We may assume
a- ^ 0. y ^ 0. and v + y ^0. We have y/x G & or x/y G (f. If v/.v G &. then
(.v +y)/x G ^. Furthermore, if .v +y <£ m. then (.v +>■)"' € $\ and hence. a--1 G 0.
This contradicts the choice of x. Hence, x + y G m. One can treat the case x/y G &
in a similar fashion. For a G & and .v g m. [ax)~x e if implies a • (ax)~] = .v_I G if.
which is a contradiction. Hence, ax G m. The above observations imply that m is
an ideal of <f. Furthermore, since «""' G & for each a 6 & - m. {(f . m) is a local
ring by Lemma 1.17.
(2) For a- G L*. either .v eif or x~[ eif. Hence L = Q[&). Suppose an element
a of L is integral over & and .v satisfies a monic relation a" + «|.v" "' + ■■■ + «„ = 0.
Ui G <f. If a ¢, &. then .v_1 g m. So 1 = -(«i a'-1 H 1- ci„x~") e m. and this is a
contradiction. Hence, a g &. We note here that m ^ (0). In fact, if m = (0). then
& = L and & is not a valuation ring.
(3) "Only if" part. Since <f is a noetherian ring, we can express m = (a-| y„).
Since a, a'1 g if or x,xj~l G &. we may assume, after a change of indices, that
a'/.v,~' G (f whenever i < /. Then m = x\tf. so we have only to put t = x\. Let
J = n„>om"- Since m" = t"(f. we have a G J if and only if t"\a for each n > 0.
Thence. J = raJ. Then J = (0) by Nakayama's lemma (see Lemma 1.28 below).
•if" part. Suppose m = t& and n„>om" = (°)- Tnen for each x £0 - (0) we
can uniquely determine an integer n > 0 such that /"|a- and t"""' |.v. Set ■u(.v) = «.
For each a g L - &. we set v{.\) = —v(x "') as .v"1 eif. Setting v(0) = oo. we
can readily ascertain that the mapping v: i->ZU (oc) satisfies three properties of
the assertion (4) above. Given a nonzero, proper ideal / of &. set r = min{v(.v):
a G /}. Since / ^ (0). ;• is determined as a positive integer. For each x G I. x e f&
because v{xt~'") > 0. Take an element .y() of/ with v;(.y()) = r. Since v{t'x^x) = 0.
trx~] is an invertible element of &. Hence, f G /. So I = f(f. Since every ideal
of <f is principal (hence finitely generated). & is a noetherian ring.
(4) This part is easily verified. □
The following result is frequently referred to as Nakayama's lemma.

1. THEOREM OF Li'lROTH
17
Lemma 1.28. Let R be a noethericm ring and let J be the Jucubson radical ( = the
intersection of all maximal ideals of R). Let M be a finitely generated R-module. If
M satisfies the condition M = J M. then M = (0).
Proof. Write M = Rm\ + ■ ■ ■ + Rms. Since M = JM. one can write nij =
53/_i aijmi (aij £•/)• or equivalently (Sjj — a,j)'' (m\ m,) = (0) if it is represented
by the matrices. Let d = det(r5,; - «,,). Then dm, = 0 for each /'. Here, developing
d. we can write d — 1 + a (a G J). Hence, d is an invertible element of R. (If
dR 7^ R. there exists a maximal ideal m of R containing dR. Hence, d e m and
1 6 m because a e J C m. This is a contradiction.) Hence, m, = 0 for each i.
Namely, we have M = (0). □
Lemma 1.29. Let {R.m) be a normal local ring with K-dimR ~ 1, Then R is a
discrete valuation ring of Q{R).
Proof. If suffices to show that for an element j.-*-1 of Q{R) {x.y e R - (0)).
either yx~' e R or xy~' e R. If x ¢. m. then yx~' e R. Similarly, xy~' e R if V ¢. m.
Hence, we have only to consider the case x. y e m. We shall show that the hypothesis
yx~l ¢. R and xy~l £ R leads to a contradiction. Let I = {a e R: a{xy~l) e R}.
Then / is a proper ideal of R and I ^= (0) because .v e /. Consider the prime divisor
decomposition \fl = pi n • ■ • np,. Then p, C m(for each i). Since K-dim7? = 1.
we have p, = m for each ;'. Hence \/l = m. So I D m" for some n > 0. Let n be the
smallest positive integer such that / Dm". Then m" • {yx~l) C R by the choice of n.
More precisely, we have m" • (>'.v_1) C m. In fact, if we assume m" • {yx~l) % m. then
there is some w E m" with iv(yx~l) = u e R — m. and accordingly a-j/~' = ii,'M_l e R
which contradicts the hypothesis. Now fix an element u G m"'~'. and let b - u{yx~l).
Then hm C m. If we write m = Rz\ + ■ ■ ■ + R:,. then bz, = Yl'- \ c<izi (cn e ^)
for each /'. Let d = det(/)<5// - (■,■/)■ Then Jr, = 0 for each j. Since R is an integral
domain and m ^ (0). we have d = 0. Expanding J. we know that /? is integral over R.
Meanwhile, since R is integrally closed, we have b £ R. Hence. m"~' • (v.v-1) C R.
This contradicts the choice of n, □
In order to discuss further relationships between normal rings and discrete
valuation rings, we need more definitions and preparations.
Definition 1.30. Let p be a prime ideal of a ring R. The height of p (denoted by
htp) is the supremum of the length h of a prime ideal sequence po ^ pi ^ • • • ^ p/, = p
which terminates at p. By virtue of the one-to-one correspondence £} >—> q = {.v £ R:
x/\ 6 £}}. q ^^ qRq between prime ideals of Rp and prime ideals of R contained in
p. we have htp = K-dim/?p. When R is a noetherian ring and I is an ideal of R.
we define the height of/ by ht/ = infp htp. where p ranges over all prime divisors
of/.
The following results will be used in a proof of Lemma 1.32.
Lemma 1.3 i (Lemma of Artin-Rees). Let R be a noetherian ring, and let a be
its ideal. Then we have the following:
(1) Let I. J be ideals of R. Then there exists an integer r > 0 such that a'"I DJ =
a'"~r{arI C\J) for each m > r.
(2) Let L = n„>0 «"■ Then aL = L.

I
IS I. PRHLIMINARIF.S
Proof. (1) Write a — («| a„). Given n variables .V| _v„. we denote the
polynomial ring R\x\ _v„] by A. For an integer m > 0. define
/ is a homogeneous polynomial of degree m and)
fUn an)ea"'IC\J j
and 5 = (Jm>u ^m- Let a be the ideal of A generated by elements of S. Then a is a
homogeneous ideal (cf. Problem II.3.1). Namely, f £ a implies that any homogeneous
part /„, of / is an element of a. Since A is a noetherum ring, a is generated by
finitely many elements f \ f, which we may assume to be homogeneous. Let
ilj = deg /',. and let r = max{</,■: 1 < / < t}. Take m so that m > r. If a £ a'"I C\J
there exists an element /(.*) .\-„) of S„, such that a = f[u\ a„). Since f £ a.
f ~ /i#i + • ■ ■ — ftgt- where #,• is a homogeneous polynomial of degree m — d,.
Hence, we have
i
a =./("[ "«) = 51//("l »//)<?/("1 a„)
!
e Y, a""''' («»'''/ n J) C a"'"' (a'7 n J).
/-1
So a'"I nJ c o'"~''(a'7 n7). The other inclusion is apparent.
(2) Set / - R and 7 = Z, in (1) above. Then a'" n L = a'" ' (a' n L) if m > r.
Hence. Z, = a'"~''L. Now take r + 1 as m. D
Lkmma 1.32. (1) Let R be a noetherian domain, let a be a nonzero element of R
with uR f R. and let p he a prime divisor ofaR, Let p~] -{e'e Q{R): sP ^ R\-
Then p ' is an R-module and R ^ p_l.
(2) Let (R. m) be a noetherian local domain such that m 7= (0) and m • m_l = R.
Then R is a discrete valuation ring.
Proof, (1) By Theorem 1.23. we can write p = (aR : b) for some b £ R. Then
ba~l £ p^1. If />£/ ' £ R. then b = ac for some c £ R. and p £ R which is a
contradiction. So ba~x ¢ 7?. Hence. R ^ p^1.
(2) Let J = n,/>om"- ^ Lemma 1.31. we have m./ = J. By Nakayama's
lemma (Lemma 1.28). we know ./ -= (0). Hence, m f nr. Let t be an element of
m - nr. Then ?m-1 C m • m^1 — R. If fm_1 ■£ R. then /m_1 C m. and hence.
tR - tm~l ■ m c m2. which contradicts the choice of t. So. /m_1 = R. Therefore.
tR = /m_l • m = m, We shall show that R is a valuation ring. Let £ £ Q[R)- and
write i! — v.v-1 with .v. y £ R. Write _v = t' u. y = tsv {r. s > 0: u. v are invertible
elements of R). Us > r. then £, = f—' (u~ '«) £ i?. If s < /■. c ' - /' ~( (»?> ')€/?.
Hence. R is a valuation ring. Now R is a DVR by virtue of Lemma 1.27. □
The following result shows that there exists a significant relationship between
normal rings and discrete valuation rings.
Theorem 1.33. Let R be a normal ring. Then we have the following:
(1) For a nonzero principal ideal aR(=f R). any prime divisor has height 1.
(2) If p is a prime ideal of height 1. then Rp is a discrete valuation ring ofQ{R).
(3) If p ranges over all prune ideals of height 1. then R =■ f] Rv as subrings of
Q(R).

I. THFOREM ()1 LUROTH
19
(4) If K-dim R = 1. then any valuation ring cf of Q{R) containing R can he
written as cf = Rp. where p is a prime ideal of height 1. In particular, Cf is a DVR.
Proof. (1) if p is a prime divisor of aR. then pRp is a prime divisor of a Rp.
(Indeed. (aR : h)Rp = (aRp : h).) Hence, replacing R by Rp. we may assume that
{R.p) is a local ring. Here we have R C p-1 by Lemma 1.32 (1). If p ■ p ' = R
holds the assertion (2) of the same lemma implies that R is a DVR. Hence, htp = 1.
Suppose p • p_1 7^ R. Then p • p_1 C p since p • p~' C R, Thence, we know that
every element £ £ p_l is integral over R (cf. the proof of Lemma 1.29). Since R is
a normal ring, we have £ £ R. Hence, p-1 = R which is a contradiction.
(2) Rp is a DVR by Lemma 1.29.
(3) Let D = p| Rp. Then R C D apparently. We shall show the inclusion D C R.
Let ceD and write t — j.v~! (x.y £ R). For every prime ideal p of height 1 we
have j.y""1 e Rv. So y £ xRp n 7?. If .v is an invertible element then c € R. So we
may assume xR ^ R. Let xR = qi fl • • ■ n q„ be the shortest expression by primary
ideals of xR. and let p, = ^/qj. Then p, has height 1 by (1) above. Hence, p, ^ p,
and q, ^ q;- whenever ;' ^= /. So for q = q, and p = p,. we have xRp = qRp and
qRv C\R=q. Hence, y e (*)/'-1 (A'^p, n R) = fl',' i 1, = v^- and £ € ^ Consequently,
we know D = R.
(4) Let m be the maximal ideal of cf. and let p = m n R. Since R c ^. we have
p ^ (0). In fact, if p = (0). then R - (0) C cf - m. and hence. 2(7?) C cf. This is a
contradiction. Since K-dim7? = 1. we have htp = 1. Hence. Rp is a DVR of Q{R).
Rp C cf. and mflfip = p7?p. (We then say that if dominates Rp and express it as
& > ^V) If both cf and 7?p are valuation rings, then & = Rp. In fact, if c e^-fip.
then £"' £ p7?p C m which is absurd. D
The next result is also indispensable in the study of algebraic varieties.
Thf.orkm 1.34. Let A he a finitely generated algebra domain over a field k. let
L = Q(A). let M/L he a finite algebraic field extension, and let B he the integral
closure of A in M. Then B is a finitely generated A-module. Consequently. B is a
finitely generated k-algehra domain.
Proof. The proof is given only in the case where the characteristic of k is zero.
With r = diva A. A is an integral extension of a polynomial ring k[x\ v,] in ;•
variables. Moreover. M is a finite algebraic extension of k{x\ v,). and B is the
integral closure of A:[.V| .v, ] in M. So. considering k[x\ v, ] instead of A if
necessary, we may assume that A is a normal ring (cf. Problem 1.1.3). Furthermore,
let M be the smallest Galois extension of L containing M. and let B be the integral
closure of A in M. Then B is a subring of B. If B is a finitely generated A-
module. then so is B because A is a noetherian ring. We may therefore assume
that M/L is a Galois extension. Since M is a simple extension of L. we may assume
that M = L(0) and 0 is integral over A. Then 0 £ B. Let n = [M : L] and
Gal(A//Z,) = {(T| a,,}. Express an arbitrary element b of B as b = Yl'!-o a>^'
(a,- £ L). Then aj{b) = Yl'i~o "/117/(#))'• Taking the matrix representation, we
have '{(7\{b) o„(b)) = E-'(a0 Q„-i). where £ = ("V W~')k, ,<„■ Let
d = det£\ Then d = 11,^/(^/(^) ~ n^0)) f ° and a'M) = ±d for each i. Hence.
d2 £ L. On the other hand. (7-,(0) is integral over A because 0 is integral over A
and (Ji(0) is conjugate to 0. Namely. Oj(0) £ B for each /'. So d2 £ L n #. Since
^ is a normal ring, we have J2 £ A. Let £' be the cofactor matrix of E. Then

20
I PRELIMINARIES
'(a,, aH_,) = d ( •'£'•'(a,(b) an{b)). Hence, a,■_, = 2,(4,/^,-(/0
(dijEB). Letc = d2. ■dndletcj = Y^jddijrji(b). Thenr, e5 because¢/,, eif.
Moreover, r, £ Lr\B = A because r, = ca,_,. Hence. /) = H"m) Q<^' G H"m) a ' (^ A')-
Since # is a submodule of a finitely generated ,4-module 2/'7o <4 ■ [0'/c). B is
therefore a finitely generated /1-module. □
We shall now state and prove Liiroth's theorem, which is the title of this chapter,
and its analogues in the theory of algebras. The formulation of Liiroth's theorem is
taken from Nagata [7]. We shall begin with a necessary general result.
Lemma 1.35. Let k he afield, let x he an element algebraically independent over
k. and let y = f{x)/g{x) be an element of k(x) not in k, where f, g are mutually
prime elements of k\x]. Consider a polynomial ring k{y)\X] over the field k(y), where
X is a variable. Then an element g{X)y — f[X) is an irreducible element of k{y)[X\.
Hence, a field extension k(x)/k(y) has degree equal to max(dcg /. degg).
Proof. Since X and y are algebraically independent over k. we can consider a
polynomial ring k[X. y] in two variables. The polynomial g{X)y — f(X) is irreducible
as an element of k{X~)\v] because it is linear in y. and it is a primitive polynomial
of k[X.y] because gcd{f(X).g{X)) — 1. Hence, g(X)y - f{X) is an irreducible
element ofk[X.y]. (This follows from the arguments used to show that a polynomial
ring R[y] in one variable over a unique factorization domain R is also a unique
factorization domain.) lfg{X)y - f(X) is reducible as an element of ^(v)^]. we can
find elements F(y.X).G{v. X) of k\y. X] (degA- F > 0. deg^ G > 0) and an element
p(y) ofk\y] such that p(y)(g{X)y - f[X)) - F{y. X)G{y. X). Since g(X)y - f{X)
is an irreducible element of k\y. X]. either F\p{y) or G\p{y). But this is absurd. □
Theorem 1.36 (Liiroth's theorem). Let k be an infinite field and let k(x\ xr)
he a purely transcendental extension of k in r variables. If an intermediate field L of
k(.\\ x,)/k has tr. deg/v. L = 1. then L/k is a purely transcendental extension.
Namely, there exists a transcendental element t of L over k such that L = k{t).
Proof. P'irst of all. consider the case r = 1. Let x = x\. Since x is algebraic
over L by the hypothesis, we write the minimal polynomial of x over L as
A (JO =X"-rf|A'"~f +--- + 0,- a £L.
Since r, e h(.x). multiplying an element of k[x] to both sides, we may write
h'{X.x) - d„{x)X" y d,{x)X" ' 4 ••• \-d„(x).
di{x) £k[x]. gcd(</o d„) = 1.
Since .v is algebraically independent over k. c, ^ k for some i. Let c — r,. and
write c — f{x)/g(.\) with f. g fe k[x] and gcd{f.g) — 1. Then g\do and f\d-,.
By Lemma 1.35. f{X) — cg{X) is an irreducible polynomial in X over k{c). and
m :— max(deg/.degg) = \k\x) : k(c)]. On the other hand, since k{c) CiC k(x).
we have
flX) - cg[X) = h(X)q(X) with q{X) e L[X\.
There exists, therefore. tp{x) € k[x] and q'(X.x) e k[x. X] such that
(*) <p(x)(f{X)g(x)-f(x)g(X)) = h'(X.x)q'(X.x).
Since the coefficients d${x)..... d„{x) of h'(X, x) have no common factors, we know

1. THEOREM OF LUROTH
21
thatip(x)\q'(X.x). Hence, we may assume y{x) = 1. In the polynomial equation (*)
in x and X. degv(left side) = m and degx (right side) > max(deg, d,-: 0 < i < n) > m.
Hence, degv(right side) = m. and consequently q'{X.x) e k[X]. If q'{X) ¢. k. there
is a root a of q'{X) = 0 in an algebraic closure k of k. Then the equation (*)
yields f(a)g{x) - f(x)g(a) = 0. where either f(a) ^ 0 or g{a) ^ 0 because
gcd(/.#) = 1. (In fact, if f(a) = g{a) = 0. then gcd{f.g) ^ 1 in k[x]. By the
Euclidean algorithm, we have gcd(/.g) G k[x] which is a contradiction.) If g(a) ^ 0.
then f{x)/g(x) = f{a)/g(a) e k. Hence, c is algebraic over k. and this is absurd.
If f{a) 7^ 0. the same argument implies c~l £ k. and this is a contradiction. So
q'(x) € k. Then f{X) - cg(X) = {d(){x)/g{x))h{X). By noting d0/g € k[x] and
comparing the coefficients of both sides of this equation, we know that dQ/g e k(c)
and h(X) e k{c)[X\. This implies that [k(x) : k(c)] = [k(x) : L]. Since k(c) C L,
we conclude that k(e) = L. So L is a purely transcendental extension of k.
Consider next the case r > 1. Since tr. degA, L = 1. we may assume that
one of x\ xr, say x\, is algebraically independent over L. Then L(x\) is an
intermediate field of a purely transcendental extension k(x\)(x2 x,■) of k(x\).
and tr.degA.( )L(xi) =,1. Hence, by induction on r, we may assume that L(x\) =
k(x\)(t) = k(t)(x\). Then we conclude L = k(t) (/c-isomorphism) by the following
lemma. □
Lemma 1.37. Let k be an infinite field, and let L\. L2 be finitely generated field
extensions over k. If L\(x) = Li(x) with an element x algebraically independent over
L\ and Lj. then L\ is k-isomorphie to Lt.
For a proof, we refer to Nagata [7, Lemma 3.12.3],
Liiroth's theorem holds ture without the assumption that k is an infinite field.
The following problem is called a problem of Liiroth type.
Let k be a field, and let L be an intermediate field of a purely transcendental
extension k{x\ xr)/k. Is L a purely transcendental extension of kl
If k is an algebraically closed field of characteristic zero and tr. degA. L = 2. then
L/k is a purely transcendental extension (A theorem of Zariski). There is. however,
a counterexample if the characteristic of k is positive and tr. degA L = 2. In case
tr. degA L > 3. the problem is negatively answered even when the characteristic of k
is zero.
The next problem is called a problem of Zariski type.
Let k be a field, and let L\. L2 be finitely generated extensions of k. If
L\{x\ x„) is ^-isomorphic to Lj{x\ x„) for variables x\ ,y„. is L\ k-
isomorphic to L-P.
Concerning this last problem, both positive and negative results are known. By
virtue of Liiroth's theorem. L\ — Li (^-isomorphism) provided L\/k is a purely
transcendental extension and tr. degA. L\ = 1. One of the counterexamples is obtained
in the case where k is an algebraically closed field of characteristic zero, L\/k is a
purely transcendental extension, tr. deg/v. Li = 3. and n = 3.
An analogue of Theorem 1.36 in the theory of algebras is the following:
Theorem 1.38. Let k be an algebraically closed field, and let k[x\ .v,.] be a
polynomial ring in r variables. Let A be a k-subalgebra ofk[x\ x,] such that A is
integrally closed and tr. deg/v. Q{A) = 1. Then A is a polynomial ring in one variable
over k.
We have to prepare two auxiliary results to prove this theorem.

I. I'RFLIMINARIFS
Lemma 1.39. Let k he an algebraically closed field, and let A be a k-subalgebra of
a polynomial ring k[x\ xr] with tr. deg/c Q{A) — 1. Then A is a finitely generated
k-algebra.
Proof. First of all. consider the case /=1. Write a- = x\. Let / be an element of
A - k. Then a- is integral over k[f ]. Hence k[.\] is a finitely generated k[f ]-module
(Lemma 1.10). Since k[f] C A C k[x]. A is a finitely generated k[f ]-module as
well. Hence. A is a finitely generated /c-algebra.
Next consider the case r > 1. We shall show by induction on r that there exists a
surjective k-algebra homomorphism ip : k[x\ a,] —> k[x] such that its restriction
ipa on A yields an isomorphism of A and <p(A). If this is done, we can identity A
with a /:-subalgebra ip{A) of k[x]. and we are reduced to the first case.
If A c k\x\ v,._i]. then A is a finitely generated ^-algebra by the induction
hypothesis. Suppose A ¢ Ar[.vi x,-\]. Let 9Jt = (a'i xr) be the maximal ideal
of k[x] \>]. and let m = A n SDT. Since k C Ajm C k[x\ x,]/Wl = k. m
is a maximal ideal of A. Consider a prime ideal (.v" — a,) (n = 1.2....)- Then
p := (a" — a,) n A C m for some n > 0. In fact, if we choose f £ A — k so that
/ ^ A:[a-i a>_i]. then we can identify / (modp) with f{.\\ xr-\. a"). So we
have only to choose n so that f(x\ a,._i . x") ¢. k. We shall then show that p = (0).
Suppose p ^ (0). Choose elements y. z of A so that >■ € m — p and :£p-(0). Then
y and r are algebraically dependent over k because tr. deg/v. Q{A) = 1. Hence, there
exists an irreducible polynomial f{Y.Z) in k[Y.Z] such that /(>■-) = 0. Write
f {}-.:) = ys + a\y'~x + ■ ■ ■ +u, + :g(y.z) = 0 (a, € /:). Since fc is an algebraically
closed field, v* + a\y*~[ + ■ ■ ■ + «, = TJ/- i(>' _ Q') (a/ e ^)- Since r e p. we have
n)-i (}' ~~ ai) € P- Accordingly, y — a,• 6 p for some /. Since x> ^ p. we know a, ^ 0.
Meanwhile. _y(0 0) = 0 if we put a^ = ■• • = a, = 0. Similarly. ui(0 0) = 0
for every element w of p. However, this is a contradiction. So p = (0). Now define
a homomorphism i//: k[x\ xr] —> k[x\ -^1--1] by y/{xj) = a,(1 < i < r — \)
and ^(a,.) = .v]'. Then the restriction yA of y/ onto A is injective. Since we may then
identify A with a Ar-subalgebra of k[x\ a-,._i]. the induction hypothesis guarantees
the existence of a surjective homomorphism ip: k[x\ x, ] —> k[x] as required. D
Take A as stated in Theorem 1.38. Then A is a finitely generated ^-algebra of
dimension 1 and a normal ring. By Theorem 1.36. Q{A) is a purely transcendental
extension k(t) over k. We shall prove the following:
Lemma 1.40. Let k be an algebraically closed field. Then the following assertions
hold:
(1) Let k(t) be a purely transcendental extension of dimension 1 over k. Then there
is a one-to-one correspondence between the set of valuation rings of k(t) containing k
and the set k U (00).
(2) A finitely generated, normal domain A over k with dim .4 = 1 and Q{A) = k(t)
is written as a quotient ring A:[a\/(.v)_i] (/(a) G k[x]) of a polynomial ring k[x].
Proof. (1) By (4) of Theorem 1.33. a valuation ring of k(t) is a DVR. Let &
be a DVR of k(t) containing k. and let m be its maximal ideal. Suppose t e &.
Noting that k[t] C &. let p = m n k[t]. Then p is a maximal ideal of k[t]. In fact,
p = (0) otherwise: whence. k(t) C &. This is a contradiction. Then k[t]/p is a finitely
generated fc-algebra domain and a field. By Theorem 1.15 and (3) of Lemma 1.10.
k[t]/p is a finite algebraic extension of k. Since k is an algebraically closed field.
k[t]/p = k. Hence, p — (t - a) with a e k. By (4) of Theorem 1.33. we have

1. THEOREM OF LUROTH
23
@ = k[t](,-<>)- Suppose t ¢0. Then t~] G m. £[/"'] C & and p := mn £[/"'] = (/"').
In this case. ^ corresponds to /_1 = 0 (considered as t = oo) and & = &[/"'](, i(.
(2) Let v'i » constitute a system of generators of a £-algebra ^. Then ^ =
k\}'\ }'A and y, = /,-(/)/£,(?)- (/,.£,- e k[t]. gcd(/,.£,) = 1(1 < / < 0) because
2(/1) = £(/). Here we shall show that there are finitely many valuation rings {&. m)
of k(t) such that g,-(t) G m for some ;'. If <^ is written as & = k[t\t_n) (for some
a G £). then #,-(/) G m if and only if gi(a) = 0. Since there are finitely many roots of
11/-i #/(0 = 0- there are finitely many valuation rings corresponding to these roots
together with @x := k[t~\,-\). Hence. A C & for any valuation ring 3'. except for
finitely many valuation rings.
We shall show that given an arbitrary element y of A - k. there exists a valuation
ring & of k{t) such that y £ 3'. Write y = f{t)/g{t) {f.g G k[t]. gcd(/.g) = 1).
Suppose g ¢. k. Let a be an element of k with g(a) = 0. and let &n = k[t\t_ay If
j' G^,». then v is written as>- = a{t)/b(t) with a. b ek[t]. gcd(a.b) = 1 and/? (a) f 0.
Moreover. f(t)h(t) = u(t)g(t). where f(a)h(a) ^ 0 and a(a)g{a) = 0. This is a
contradiction. Hence, y £ &n. Suppose g G k. Then y = g~x{f{u~x)ud/u(l). where
u = /^1. d = deg, /'. and f(u~])u(l e £[//]. Hence, y ¢. $^. Therefore, there exists
a valuation ring & with A (fl &. If & = &n with a <E k. then we may assume ^1 = &x
by replacing ? by (/ - a)-1. Let &a> (a,- G £: 1 < / < n) exhaust valuation rings
of £(/) (except for &x) such that A ¢. &ai. We have shown that there are finitely
many of such valuation rings. Let /(/) = Il;'-i(? ~ a')- By Theorem 1.33. there
is a one-to-one correspondence between the set {p; a prime ideal of A of height 1}
and the set {&: a valuation ring of k(t) with & D A}, and we have A = (~) Ap.
Consequently, we have
A=f]Ap= f]ff= f]ff„=k[t.f(t)-1]. □
p *D4 net.
<* /a,
Proof ok Theorem 1.38. It suffices to prove the following result:
Let A be a £-subalgebra of a polynomial ring £[aj .v,]. and let
A* = {invertible element of A}. Then A* = k* = k - (0).
This follows readily from the fact A* C k[x\ v,]* = k*. Since A =
^-./(0-1]- ^* § <4* provided /(?) ^ k. This is a contradiction. So. f{t) e /c
and A = k[t]. D
1.1. Problems
1. Show that k' <8>a £(£i ¢,-) is a field if k1 /k is a purely inseparable extension
and ¢1 ¢, are elements transcendental over k.
2. Show that if k'/k is a purely inseparable extension and /(.v) G £[-v] is a separable,
irreducible polynomial, then f(x) is also a separable irreducible polynomial as an
element of k'[x]. Here we say that f(x) is a separable polynomial when f(x) = 0
has no multiple roots in an algebraic closure of k.
3. Verify that a unique factorization domain is an integrally closed, integral domain.
A polynomial ring k[x\ v„] over a field is. in particular, an integrally closed.
integral domain. Verify also that a quotient ring Rs of an integrally closed,
integral domain R with respect to a multiplicatively closed subset S is an integrally
closed, integral domain.

24
i. PRELIMINARIES
4. Let R be a noetherian domain. Show that a necessary and sufficient condition
for R to be a normal ring is that Rm is a normal ring for every maximal ideal
m of R. (Hint: For d; e Q{R) which is integral over R. show that the ideal
I = {a e R;ac e R} equals to R.)
5. Show that in a polynomial ring k[x\ x„]. a prime ideal next big to (0) is a
principal ideal.
6. For an ideal I = (x2.xy) of a polynomial ring R = k[x. y] in two variables,
prove the following results.
(1)/ = (.y) n (x2.y) is a decomposition of I into primary ideals. Hence, (x)
and (a. v) exhaust all prime divisors of /. Here {x.y) is an embedded prime
divisor of/ because (.v) ^ (-v.j).
(2)(x) = (I :x+y) and (x.y) = (1: x).
7. Show that A/m is a finite algebraic extension of k if k is a field. A is a finitely
generated Ar-algebra. and m is a maximal ideal of A. Show that if k is an
algebraically closed field, then every maximal ideal of a polynomial ring k[x\ x„]
is expressed as (x\ — <x\ .y„ — a„) (a,- £ k). Show that there is a one-to-one
correspondence between the set of maximal ideals of k[x\ x„] and the set
k" = {(a\ a„): a, G k\.
8. Let k be an algebraically closed field, let k[x] be a polynomial ring in one variable,
let f(x) be a polynomial of degree n without multiple factors, and let A =
k[x.f{x)~]]. Show that k* is a subgroup of A* and A*/k* is a free abelian
group of rank n. Here A* is an abelian group with respect to the multiplication
in A and is called the multiplicative group of A.
9. Verify the following assertions.
(1) Let (/4.m) be a noetherian local ring, and let r = K-dim^. Then there
exist elements u\ w, of m such that the ideal (u\ «,.) is a primary ideal
belonging to m. Then {u\ m,.} is called a system of parameters of A. (Use
the following Altitude Theorem of Krull (cf Lemmas II.4.8 and II.4.9): Let R be
a noetherian ring, and let a be an ideal generated by r elements. Each minimal
prime divisor p of a then has ht p < r. For a proof of the Altitude Theorem, see
Nagata [8, p. 26].)
(2) Let 5 be a finitely generated algebra over a field k. let m be a maximal ideal,
and let A = Sm. Then K-dim^ = tr. deg,, Q(S).
(3) Let tp: (A.m) -» (B.n) be a homomorphism of noetherian local rings such
that ¥>~'(n) Q in. which is then called a local homomorphism. Then we have
K- dim B < K- dim A + K- dim(fl/mfl)
10. With the notations in problem 9 above, assume that B is a flat ^-module. (Namely.
for an injective homomorphism of ^-modules f: P —> Q. f ($A B: P &A B —>
Q (8>,.) B is also an injective homomorphism.) Then the equality holds in (3) of
problem 9.
11. Let R be a principal ideal domain, and let M be an R-module. Show that M
is a flat R-module if and only if M is a torsion-free /^-module.

CHAPTER 2
Theory of Sheaves and Cohomologies
When we treat manifolds, including algebraic varieties to be discussed in the
present book, we often discuss global and local properties combined together. There,
central roles are played by cohomologies with coefficients in sheaves. In this chapter.
we discuss cohomology theories on ringed spaces.
Definition 2.1. Let / be a directed set. An inductive system (or I-inductive
system) of sets indexed by / consists of sets and mappings & = {S, (i e /)./;,■: S, —►
Sj (i < ./)} satisfying the properties: />, = fkj ■ //7 whenever i < j < k. and
/,/ = id.s- for each i e I. Let & = {S,' (/ € /).' /J-,: Sj -> S) (i < /)} be
another /-inductive system of sets. A collection of mappings {t, : S,- —> 5,': i € /}
is called a morphism of /-inductive systems if /J, • t, = t; • /'/, whenever ;' < ./'. We
abbreviate an /-inductive system 6 by (S,),6/ and a morphism of/-inductive systems
{t, : S,- -> Sj:; € /} by (t,),g/ : (S,),e/ -^ (S,'),e/. Let 6 = (5,),e/ be an /-inductive
system. Given a set 5 and a collection of mappings [a,•: S; —> 5; / e /}. 5 is called
an inductive limit of 6 (denoted by S = lim S, if the following two conditions are
satisfied:
(i) (Tj • fji = a-, whenever ;' < /. Namely. (fT,)/e/: (S,-),-e/ —> 5' is a morphism
of /-inductive systems, where £ denotes an /-inductive system S-, = S. /;, = ids
(' < J)-
(ii) If (T,),e/: (S,-),-e/ —» 7" is a morphism of /-inductive systems, then there
exists uniquely a mapping of sets p: S —> T such that t, = p • a,- for each i e /.
In the above definition, if we replace sets by abelian groups and mappings of
sets by homomorphisms of abelian groups, then we can define /-inductive systems
of abelian groups and their inductive limits in the same fashion. If we consider a ring
R. R-modules and homomorphisms of /^-modules, we can also define /-inductive
systems of R-modules and their inductive limits.
Lemma 2.2. Let & = (S,),e/ be an I-inductive system of sets {or rings, or R-
modules. resp.) indexed by a directed set I. Then there exists cm inductive limit lim 5,
which is a set (or a ring, or cm R-module. resp.).
Proof. We consider each case separately. When G is an inductive system of
sets, introduce an equivalence relation on a direct sum S' = JJ, 5, as follows: For
,v,- G Sj. Sj e Sr s, ~ .v, if for some k € I.i < k.j < k. /a,(.v,) = /a/(.v/). Let S
be the set of equivalence classes S'/ ~. and let rr,: S, —> S be a composite of the
natural embedding S, into S' and the residue mapping (.v' h^ the equivalence class
of .v'). Then {rr,: S; —> S; i E 1} is an inductive limit. Consider the case where &
is an inductive system of rings. Define the sum and the product of two elements
rri(x).m(y) of S(x € S,.y G Sj) by mix) + m(y) - rrk(fki(x) +/,„(>•)) and

2f>
1. PRKLIMINARH'S
a,\x)a fy) = cTk{fii{x)fki(y))- where / < k.j < k. We can readily ascertain that
this definition is independent of the choice of representatives .v. v of the equivalence
classes a,(a). a-,{y) and the choice of an index k. In this way. S is a ring with
respect to the above-defined sum and product. The mapping u,: S; —> S is then a
ring homomorphism. S is an inductive limit of & as rings. When & is an inductive
system of /^-modules, we have only to define the sum and the scalar product by an
element of R on S as in the case of rings: <r,-(.v) + rrfy) = crk(fki(x) + /a/0'))-
arr,[x) - t7;(ax)(a £ R). □
Consider anew an /-inductive system of commutative rings 3 — {Aj {i £ /).
ap: Aj —> A, (i < /}}. An /-inductive system of modules 9Jt :- {M,- (/ e
/)../'//: M; —» M/(f < j)} is. by definition, compatible with 21 if A/,- is an -4,-module
for all/ and fp{a,m,) - «,,(«/)/ ,,(«7,0 foreach a, 6/4,- and for each m,- e M, (i <j).
We say simply that 97t is an /-inductive system of'^-modules. Let Wl' ~ {Mj. /J,} be
another /-inductive system of 2l-modules. A homomorphism of/-inductive systems
of 2l-modules (tp, ),<=(: 3JT' —^ OT is. by definition, a collection of /1,-module homomor-
phisms <p/: M/ —* M, (for all;' e /) such that </>/("//(«,■).//,('«,')) = «//(«,•)///•¥>/('",')
(«,- € A^m'j G A//) whenever / < /. With a homomorphism of/-inductive systems
of 2l-modules ((//,),-/ •' 5OT -> 2Jt" =- {A//'./",}, we say that a sequence
o -> an' ^1 an ^1 <m" - o
is an exact sequence of I -inductive systems of 2l-modules if
o — m; --> m, --» M," — o
is an exact sequence of .4,-modules for each i.
LiiMMA 2.3. Let
0 -4 W (A' 9Jt ^ OT" -+ 0
be an exact sequence of I-inductive systems of^-modules. Then the natural sequence
of Urn Aj modules
0 — lim Mj ^ lim M, "> lim M," -- 0
f'.v exact.
Proof, An element ; of lim M" is represented by r,- £ A/," for some /' and
^/,: Mj —» A//' is surjective. Hence, (// is surjective. Suppose y/{y) = 0 for_y £ Hm M,,
where y is represented by y, £ Mj for some /'. Since /",(yv(j',)) = 0 for some / > /
and /'',(y,(y,)) = ^/(//(j7;))- there exists a-/ £ M\ such that /;,■(>';) = ¥>,(*,).
Let a £ lim M\ be represented by v/. Then y = <p(x). Hence. Ker yj = Iniv?.
Suppose tp(x) = 0 for v G lim A//. Let _y,- £ A/,' represent a. Since f j,{fi{x,)) =
V/(//A-Y')) = 0 f°r some 7 > ' ar>d V7/ 's injective. we have f'fixj) = 0. Thence,
v = 0. and ^ is injective. □
Let X be a topological space. We define a presheaf of sets on A' as follows.
Definition 2.4. A presheaf of sets & on a space A' assigns a set &>(U) to each
open set U of X and a mapping of sets pvv : 3?>{U) —> 3?>{V). called the restriction
morphism. to each inclusion of open sets t7 C t/. subject to the conditions: For open

2. THKORY OK SHLAVLS AND COHOMOLOGHS
27
sets W C V C U it holds that /^y;. = /^r • p\-f. and also. /^r = id^-j for all
U. We denote pw also by />f-..
In the above definition, the set 9(%) is usually not assigned for the empty set 0.
Instead of assigning sets and mappings of sets, we can consider abelian groups 9(U)
(or commutative rings, or modules, resp.) and homomorphisms pyu\ 9{U) —>
9(V) of abelian groups (or commutative rings, or modules, resp.) to define a
presheaf of abelian groups (or commutative rings, or modules, resp.). We then take
9($) to be (0). We call a presheaf of abelian groups simply an abelian presheaf.
Let 9. (S be presheaves of sets on X. We say that a morphism (or a homomor-
phism) of presheaves u: 9 -+ & is given if a mapping of sets of u(U): 9(U) -+@(U)
corresponds to each open set U and the equality piL • u{U) = u(V) -pre holds for
any inclusion of open sets V C U. Here we have used the same notation p\c to denote
the restriction morphisms of 3P and (§. We can define a composite v • u: 3P -+ 91 of
morphisms of presheaves u: 9 -+ (S and v: € -+ 91. Considering abelian presheaves
or presheaves of commutative rings instead of presheaves of sets, we can define a
morphism of abelian presheaves or a morphism of presheaves of commutative rings.
A presheaf 9s on X is called a sheaf if it satisfies the following two conditions
(F\) and [Ft) for an arbitrary open set U of X and an arbitrary open covering
U = {Ua:a £ A} of U:
(F|) If for x.y £ t?(U) pa(x) = p„[y) for each a e ^. then .v = y. where
Pa = /^;,t •
(F2) If for n„-v„ £ IL W«) ^(x„) = /;f/((x/() for each a.0 e A. then
*„ = A*M for some .v £ ^>(£/). where Ua/I = Un n £/,/ and ,^/( = /^,l/(t„.
We can summarize these two conditions by saying that the sequence
(F) ^(U){-^l[9(Un)^l[9>(Unll)
" ~^ a.ft
is exact where the mappings numbered (1). (2). and (3) signify, respectively, the
mappings x ^+ \\apn{x). \[a x„ >-> EL,//;«/<(*«)• and Il„ A'« ^ IL./»/#>(•*«)• Note
here that in the product f[n^9(Ua^) the components 9{Unp) and 9(11/1,,) are
distingushed from each other.
When 9 and <§ are sheaves, a morphism of presheaves u: 9 -+ (§ is called a
morphism of sheaves.
Let 9 be a presheaf on X and let x be a point of X. Let 7\ be the set of open
neighborhoods of x. Then 7\ is a directed set by defining an order in 7\ by U < V
if V c U. A collection {9(U): U £ 7\} forms an inductive system indexed by 7\ .
Its inductive limit lim 9(U). denoted by 9y. is called the stalk of 9 at .v. If 9
is an abelian presheaf or a presheaf of commutative rings the stalk 3PX has the same
algebraic structure. If u: 9 -+ € is a morphism of presheaves. then we can naturally
define a mapping of stalks ux: 3?x -+ &x.
When we treat presheaves or sheaves in the present book, we frequently consider
the following situation. Let M be a presheaf (or a sheaf) of commutative rings on
X. A presheaf (or a sheaf) of modules Jt on X is called a presheaf (or a sheaf)
of s$ -modules if .#(£/) is an .(/((7)-module for any open set U and the restriction
morphisms /v i t and ps/.vc of Jt and .»/ are compatible with each other for V C [/.
i.e.. pjr.\L{am) = psi.vi \u)pjt xi;{rn) for w e Jt{U) and a £ .9/((/). We define a
homomorphism of presheaves (or sheaves) of .af-modules f: .# -+ jV as a collection

28
I. PRELIMINARIES
of M(£/)-module homomorphisms f(U): M{U) —-+ JV{U) for each open set £/ such
that /({/) and /(K) commute with the restriction morphisms p^A-i: and pyr.ru of
^ and yf whenever V C U. Looking at the stalks at x £ X. we see at once that
J(x is an ,afv-module and /\ : y£x —> JVX is a homomorphism of j/v-modules.
A sequence of presheaves of ^/-modules
(*) >Mi-\ ^1 M; !^Jti\\ /jA>' j?n2 -> ■•■
is called an exact sequence if
(*)[■ > *i-i(U) f-4u) MAU) J^] Jti+\{U) Mn ■ ■ ■
is an exact sequence of sf{U) -modules for every open set U. The sequence of stalks
is then an exact sequence of ,&\-modules by Lemma 2.3.
For a presheaf 3P on X. we define the sheufification "3$ of 58. For this purpose,
we consider the set of stalks {2Px;x £ X} and the direct sum 3s = Uv€X ^- Define
a topology on 58 as follows. Consider a subset V(U.a) = {av:-f £ £/} of ,98 for
an open set U of X and « e .3°(U). where ax denotes the element of ^ (regarded
as a point) determined by a. When U ranges over all open sets of X and a ranges
over all elements of 3P(U). we can define a topology on 3d with {V(U.a)} as a
fundamental system of open neighborhoods. In fact, for v £ V(U.a) n V(U'.a').
there is an x e U (~) U' with «x = a[. = v. So by the definition of 8PX, there exists
an open set W of X such that x G W C U n U' and pwu(a) = pww{a')( — b).
Then F(W.fc) C K(t/.a)n V{U'.a'). Define a mapping ti: & -► X by u £ S\ i->
x £ X. Then re is clearly a continuous mapping, and for each v € 3&x c ,58. an
open neighborhood K([/. a) off is homeomorphic to an open neighborhood U of
v = n(v). So n is locally homeomorphic. A pair of 3s together with the projection
n (or simply n: 3? —> X) is called the sheafifying space of 58. For an open set C/
of J*f. a continuous mapping „v: t/ —> ^8 is called a section of n (or 3s) over C/ if
7r|[/ • ,v = idc. Define a presheaf ,^ by C/ i-> !F{U) = {.v: £/ h^ 2P; a section over
£/}. pi l\ s £ !F{U) i-> .v|r £ &(V). Then we have the following result.
Lemma 2.5. (1) ^ /,v a .s'/;ra/.' //,58 /.v a presheaf of abelian groups,
commutative rings or modules. & has the corresponding algebraic structure. (We call & the
sheafification of 3$ and denote it by "&.)
(2) Define a morphism of presheaves u: 3s —> !F by u(U): a £ &>(U) >-> sa £
&~{U). where s„ : x £ U >-> ax £ 3*. Then u has the following property: If v. 3s —> !§
is a morphism to a sheaf '2?. then there exists a unique morphism of sheaves v : EF —> *§
such that v = v • u.
(3) {"&), = 3*,. for each x £ A'.
(4) "^ = ^/»r « sheaf!?.
Proof. (1) It suffices to show that the foregoing two conditions (/*")). (Fi) hold
for y. Let C/ be an open set of A" and let iX = {Ua:a £ A} be an open covering
of U. Given .v.? £ !F{U) with />„(.y) = />„(0 for each « S ^4. then ,v(.v) = t(x) for
each x e C/. Hence, .v = r. Given .v(V £ 5"(£/„) (for all a £ ^) with -va|(,-„,, = ■v/dc^,
for each a. ft e A. define a mapping .v: C/ -^ 3d by .v|L'„ = *'„. Then s is apparently a
continuous mapping and n\L- ■ s = idL . Thereby. 9~ is a sheaf. If ^8 is. for example.

2. THI'.ORY ()(■ SHI-.AVHS AND COIIOMOLOGIKS 29
an abelian presheaf. define the sum s + t for s.te F(U) by (.v + t)(x) = s(x) +t(x)
(the sum in &>x). Then F is a sheaf of abelian groups. The same arguments apply
to the other cases.
(2) For an open set U of X define a mapping v{U): ,T{U) —> 9{U) as follows.
An element s of SF{ U) is a section of & over U. Since .v(.v) S 5PX for .v g t/. there exist
an open neighborhood Ux of v and tf(.v) <^2P{UX) such that F(C/X. c/(.v)) = \{UX).
(Note that .v(C/) is an open neighborhood of ,v(.v) in 5P.) Since {Ux:x e U} is an
open covering of U. we may assume that there exist an open covering it = {Ua:ae A}
and aa e &>(U0) for each a £ A such that K(C/„. c/0) = s(Un). Then since for each
a.p £ /1 («„)\ = («//)» = -v(.v) for each ,y e C/fV/(. there exists an open covering
*%<*/< = {^a/;,:)' e 5„/(} of Un/j such that «„|ir/( = c//;|r,i/; for each 7 e 5fv/(. Set
g„ = ?•(£/„)(«,»). Then #„ e .??(£/„) and ga|r,1/( = *■'(«« I r,„, ) = v{a/<|r,„J = ^,),-^
for each a.p € A. Since 2? is a sheaf. £«|l„/; = #/<k„/;. Hence, there exists uniquely an
element g of 9{U) such that #„ = g\Ln for each a e A. We then define v(U){s) = g.
We can thus define a morphism of presheaves v: & —> gf. The definition of j/ and
£> implies apparently that v = v ■ u.
(3) This part is immediate from the definition.
(4) This part can be proved in the same fashion as for (2) above. □
By virtue of (2) of Lemma 2.5. a morphism of presheaves w : & —» @ determines
uniquely a morphism of sheaves "w : "& —> "(§ such that u$ • w = "w • u&. where
u^: & —> "'5P and u& : € —> "(§ are the morphisms given in Lemma 2.5.
We call a sheaf of abelian groups simply an abelian sheaf. For a while, we shall
treat abelian sheaves and homomorphisms of abelian sheaves exclusively. A sequence
consisting of abelian sheaves and homomorphisms
(*) ---^9-, ,^.^^,^,^3^...
is called an exact sequence of sheaves if the sequence of stalks
is an exact sequence for each .v <E X.
We shall prove the following result.
Lemma 2.6. (1) Let p: & —> "S he a homomorphism of abelian sheaves. Define
presheaves Jf and%' by 3?{U) = Kerip{U) andW'{U) = Cokertp(U). Then Jf is
a sheaf though W is not necessarily a sheaf. With W = "W. we have the following
exact sequence of abelian sheaves
(We call 3Z and'S the kernel and the cokernel off. respectively, and write 3Z — Ker ip
andW = Cokery>.)
(2) For an exact sequence of abelian presheaves
> Jt\-\ j^] Jt\ A Jti+\ J^ JT/+2 ->■•■.
the sequence of their sheafifications
_> or. , vlz} or. 2t\ or. , ^1 <jr _^
is an exact sequence of sheaves, where &-, = "JP, and <p: = " /',.

1. RRhLIMINARIhS
Prook (1) Let U be an open set of X. and let it = {Ua;a e A} be an open
covering of U. Then we have the following commutative diagram:
0
Y
.mu«
0
■&(U)-
c(L
0
■&(u)
0
■W'{U)
^Yl^(u„)IM±lll&(uo
^X\%\u„
0
„./i
a./I
riv-tt
a./;
,,./1
/0
-*o
0.
Here three rows are exact sequences by the definition and the middle two columns
are also exact sequences by the definition of sheaf. In the sequence (F) considered in
the definition of sheaf, we may replace homomorphisms (2) and (3) from T\t ^{U„)
to nna&~{U„it) by a single homomorphism (2)-(3). Then by chasing the above
diagram, we know that the leftmost column is an exact sequence, though we cannot
conclude that the rightmost column is an exact sequence. (As problem 1.2.2 shows.
there are examples for which the right column is not an exact sequence.) Let //: "S —>
,?' be the morphism of presheaves determined naturally by the definition of??'. Then
p : = "/)': 5/ —> % : = U(S' is a morphism of sheaves. Let /: 3£ —> SF be the natural
morphism of presheaves. Then ; is a morphism of sheaves. Hence, we obtain a
sequence of abelian sheaves
0
■if
J^^Ag"
0.
For each .v £ X. if U ranges over the set of open neighborhoods of .v. the sequence
of inductive limits
0^ \im5?{U) -> limbic/) -> \\m&{U) -> hmW'{U) -► 0
\€t \ef ^e^, vet'
is an exact sequence by Lemma 2.3. If we note that {"£P) v = ,_?\. we obtain an exact
sequence of stalks
0 -> X, -¾ 5\ ^ &x ^ g\ -» 0.
The sequence (*) is therefore an exact sequence.
(2) Since the inductive limit lim preserves the exactness of sequence, we
>i€(
obtain, as in the above proof, an exact sequence of stalks
• ■ ■ —> Jvt j _ | x —> yvi i x —> yjt; 4- ] v —* -^/12 \ —' • • ■ .
Noting that .?",., = (".#,)v and /,.x = ("/,)A. we know that the sequence is exact
as stated. □
Let /: X —+ Y be a continuous mapping from topological spaces X to Y. We
shall treat below only abelian presheaves and sheaves. Let & be a presheaf on X. We
define a presheaf j\2P on F by f*£P{V) = 9i{f~x{F)) and the restriction morphism
/>r'i {/,^) = P, -!{]■')/■ 1(1-)(-^)- Wecall /,^5 the <///•«? image of & by /'. Let<f be

2. I HHORY OF SHFAVFS AND COIIOMOLOGIhS
a presheaf on Y. We define a presheaf f'S on X by fS{V) = lim €(V).
t- j —>/([-)cr
where V ranges over all open sets V of Y containing f{U). The restriction morphism
for f*(§ is derived naturally from the one for (§. If 'S is a sheaf on Y. we denote
the sheafification "{/'&) by f*& and call it the inverse image of ^ by /'.
Lkmma 2.7. (1) If SF is a sheaf on X. f »SF is a sheaf on Y.
(2) IfS is a sheaf on Y. then {f*&), = %f(x) for each x e X.
(3) If & is a sheaf on X and 'S is a sheaf on Y. we have Horn,y(/"£\ .9~) =
Horn y(&. /*JF). where Hom.v( . ) or Hom><( . ) denotes the group of all homomor-
phisms of given sheaves.
Proof. (1) Let V be an open set of Y. and let 2J = {V,:k e A} be an open
covering of V. Then f~]{V) is an open set of X and /"'(23) '■= {f ~\V-,.): k € A}
is an open covering of /'_1 (V). j\,.9~ is then a sheaf because the following sequence
is exact
F{f-]iv)) ^X\^(f-](v,))^^\.T{f-'{v})n f-\vLl)).
'/ /..it
(2) By (3) of Lemma 2.5. (/*2?)v = {.!'"&),■ If V ranges over all open
neighborhoods of .v. we have
(/•g?), = lim(/*S0(t/) = lim lim *?(K).
igf \er /(c icr
where it is easy to see that the last double inductive limit is equal to lim rS{ V).
—>/(\)er
V ranging over all open neighborhoods of ./ (v).
(3) Let tp: /*2? —> f? be a homomorphism of abelian sheaves on X. Namely.
we have a homomorphism <p{U): f*&{U) —» &~{U) for every open set t/ of X such
that (p{U) and <p{U') commute with the restriction morphisms whenever U' C [/.
By the definition of sheafification. an element s of f*&{U) is determined by giving
an open covering {Un:a £ A} of t/ and ,s„ e f'&(U„) for each a e A such that
(■V>)\ — Cty)\ for each .v £ Uap:a.[i e ^. Let /„ = y>(C/(1)(.va). Since .9" is a
sheaf. ta £ &{Un) {a e /1) patch together to form t e !?{U). This f is equal to
<p(C/)(.v). Thence, giving an element y> of Hom^{f,(S.!¥) is equivalent to giving a
morphism of abelian presheaves u>': f& —> &'. Since f'&(U) ~ lim ,<^( T)
r r f ./ y —>/((')cr
for an open set U of A\ giving <p'{U): f'&(U) —> .^((7) is equivalent to giving
y/(K): rS{V) -* &-(U) for all open sets V of F with f{U) C r so that V/'(K) =
v'(^/')/,i',r whenever /(£7) C K' c F. Namely. (//'(F) is a composite of/^, / i(, ,
and y/( V): &{V) -► 3^{f'](V)). Therefore, giving <p'; /*2? -^ & is equivalent to
giving y/: 'S —> f*&~. □
A ringed space is a pair (X.srf) consisting of a topological space X and a sheaf
of commutative rings. If the stalk Mx is a local ring for each .v e X we call [X..sf) a
local ringed space and write @x instead of M. We then denote the stalk by @x v-
the maximal ideal by m.v.\ and the residue field by A (a). A morphism from a
ringed space (X.saf) to a ringed space {Y.3%) is a pair $ = (/. y) consisting of
a continuous mapping / : J*f —> F and a homomorphism <p: 38 -^ f\srf of sheaves
of commutative rings. We often write / for ¢. By Lemma 2.7. we may equivalently
give a homomorphism <p#: f%38 —> ,i/ of sheaves of commutative rings instead of

I I'RIUMINARU.S
A morphism of ringed .spaces (/-</?): iX.<$\-) —* f Y.<?y) i*- by definition, a
morphism of local ringed spaces if iVwK : (./'"'<^Y)\ = &\ /lv! — $V .< is a homomorphism
of local rings, called also a local homomorphism. i.e.. (0^),(111, ,-:,,) C mx.\- for each
,\ G A'. Here note that (,^)., is a composite of <pv: &'yv —*■ (./,(^-),- and the natural
homomorphism f/„^\), -- lim ^'-, (/' "' ()')} — lim ^\{U) - ffx v. where
r = ,/'(.v). A composite of morphisms of ringed spaces [f.ip): iX.stf) —> (K-5?)
and {g. y/): ( K/^) -► (Z.^) is (# • f.g,l.<p) • '/>■
We shall consider various sheaves of s? -modules on a ringed space (X..9). We
call a sheaf of .^-modules simply an .'9-Module. From two .^'-Modules ,c/ and %'. we
can define the following three .^-Modules 9 tr.9. :T •[■^ .§' and ^m,a {.T /9). First
of all. we define a direct sum 9' ]' .9 by [9 --^)( U) = 9 ([/) > | .?■( 1/). In the case of
a tensor product 9~,-).,.,, ,9. define a presheaf 9 by .95 f V) = 91 V) '^^lr) 9{U) which
is not necessarily a sheaf (cf. problem 1.2.3). Then we define9~ ">:,/ ?/ as ''9°. In order
to define the third sheaf, define Hom.,,^.?..?) as the set of all homomorphisms of
.^"-Modules ip: 9r —%'. Hoivij(J,.?) is an .9 (X J-module if we define a sum p h i//
by (if + i//)(U) = <p\U) + y/(U) for ^s. y/ G Hornby 9'.'^') and a scaler product ay> by
(aiys)(l/) = pi x{a) ■ tp(U) for a G s9(X). Write sn'i . 9~t . and 5?/ the sheaves on U
obtained by restricting s/.;9 and 9 onto an open set U of X. respectively, For open
sets V c_ t/. ^ e Hum-/, (5"; 99\ ) y-* <pt e llonv,\9'\ 99\-) is a homomorphism of
modules which is compatible with the restriction morphism pri_ '■ sf{U) —> ,,9/(17).
We denote this mapping by the same notation />( ; . Then {Horn./, (;9L .9L ■)■/>!'{ }
forms a presheaf of .s/-modules W on X. Now let ?/ be an open set of X and let
{Un:a c /1} be an open covering of U. If for <^.;// G Honv, (5*) .¾^ ). o?|[-„ = y/|[ri
for each a g /(. then 93 = y/ because .^ is a sheaf. Suppose ipn G Hom.^ (-5^-,, -^( ■„)
(« G /4) is given so that y=„|r,„: — V/'U , f°r eacri ^-A e -^- Since .^ is a sheaf, we
can construct p G Honv, (.9t .9L ) in such a way that <£>|( =- <pn for each a e A.
Hence, the presheaf dT is a sheaf which we denote by %?pmvJ\9~.%). In particular,
if 9 = .a/, then we define 9""' :— #%^,K? {.9.,s/) and call it the dual sheaf of .9.
Lkmma 2.8. With die above notations. [9" -.<-^ 9), = 9~x W.c/. '9X for each a g X.
Proof. We have {9~ ^ &), = lim 9>(U) = lim ^W) :<^[r)W{V) =
,'A>V, •§*■ where we use a general fact in the last equality that the inductive limit
commutes with the tensor product. U
Let $ - if.ip): [X.s£) —«■ (Y..93) be a morphism of ringed spaces. The direct
image /',.!? of an .(/-Module^ is an /Vijf-Module. We regard f ,9 as a ^-Module
via a homomorphism of sheaves of rings ip: 9% —^ f j9 and denote it. by <$>t9~. We
call it the direct image of ,9~ by <t>. If.1? is a ^-Module, then /1.^ is an f*9S-
Module. Let <J>\5' = ^' X7 • ,^ /'*,?' in view of a homomorphism of sheaves of rings
ipn : f*.93 — .&'. Then ¢^ is an j/-Module. We call this the inverse image of "§ by
¢. There is the following adjoint relation between the functors of taking the direct
image and the inverse image J i-<I),J and %' ^ <£>'%.
Lkmma 2,9. With the above notations. Horn4(<1>*W. 9~) = \\om,s(^.^>.,.9).
Proof. Giving a g Worn.,;{<&*"9.9") is equivalent to giving an
jf[U)-homomorphism a(U): 0+.%'(t/) -^ .9~(U) for each open set U which is compatible with the
restriction morphisms. n(l>) is given as follows. Namely. o[U) is represented by a
collection of homomorphisms of ..'9(/./,,.)-modules au : :9[U„) ■:•• f /!U , f "9{UIV) -^

2. IIIIORYOI SH|:AV1,S \NI) (')IK)M()L()(jlI s
tF{U(>) {a G A) such that t„|(;„;, = ^//1(,,,, for each a.fi £ A and each I e ,-1,,/().
where { (/„: a G A } is an open covering of V and { (/„//,: A G /1,,/(} is an open covering
of Ua/j for each a.fi £ A. Furthermore, giving a homomorphism of .8/( (/,, i-modules
an is equivalent to giving a homomorphism of /" .3? ((/„)-modules a'n : /'.'</((/,,) —>
SF{Ua). where we regard the ,$/((/„ )-module ,5^((/,,) as an /^((/,,)-module via a
ring homomorphism/^( [/„)—>,s/( (/,,). By the above hypothesis. a'n\i- , --- ^1(,,,,
for each ?. G /1„«. Since !F is a sheaf, it', = n'.Xi for each a. Ii G /1. Hence.
{n'/.a G A} determines a homomorphism of /'.5?((/)-modules a'(U): f''&{W) —>
!F(U). Conversely, given a homomorphism of /^((/)-modules a'(U): f"ff{\J) -*
!F{U) for every open set U which is compatible with the restriction morphisms. we
can determine a homomorphism of j/((/)-modules 17((/): <$>'&(U) —> 7{V). In
sum. Homyir^.J) = Horn/,.^(/^.^.,^). An element rr' G Horn/ ..^(./''.^.!?)
gives, by Lemma 2.7. a homomorphism of abelian sheaves r: ,r</ —> f'..9~. which
we can show easily to be a homomorphism of .^-Modules. Conversely, given r G
Hom,w(ff.O,,f) we can determine a' G Horn f ,#(/*??. ZF). D
We shall consider various conditions on ,e/-Modules. where (X..sf) is a ringed
space. For an .a/-Module SF. a homomorphism 11: ,8/ —> .9^ determines an element
,v EfF{X) as s — u(X){l). and vice versa. Let / be any index set and let ,e/l/l denote a
direct sum ®,6/ ^,. Namely, we define (0/6/ s/e,)(U) = 0,e/ ,a/((/)c,l( . where
c'i\i = Piwi'-'i)- We call {/,: i € 1} ci free basis of ,sf{,). Giving a homomorphism of
^/-modules u: ,a/(/) —> ,9~ is equivalent to giving a collection of elements {,v,: i 6 /}
of .^ (A") such that (/(A'Xc,) — ,v,. If I is a finite set of n elements, we denote ,s/(/l
by sf". Concerning the u above, ti is surjective. i.e.. Cokenc =- (0). if and only if
^ = E,t/ -<U)v for each x e X.
Dkkinition 2.10. Let .T be an .(/-Module.
(1) ,9~ is qnasicoherent if. for each x e X. there exist an open neighborhood (/
of v and an exact sequence of .of; -Modules
(.a/)1'71 -> (<#, )l/! — .Sy — 0.
where j/|( and ^"|( are the restrictions of sf and ,^ onto (/. (We also write ,»fr
and !FL). Here / and J are not necessarily finite sets.
(2) ?? is finitely generated if. for each .v G X. there exist an open neighborhood
U of v and elements ,v, (1 < / < n) of .9~(U) such that .5^ = £]" , ,a/, (.v,), for each
v G U. Here the integer n may differ depending on the point .v. We can define & to be
finitely generated equivalently by saying that there exists a surjective homomorphism
(,#t )" ^ 9-r -* 0.
(3) .T is coherent if the following two conditions (i). (ii) are satisfied:
(i) ,9~ is finitely generated.
(ii) For every open set U of X. every positive integer n and an arbitrary
homomorphism of j/(-Modules y>: (.a/; )" —> 5^ . K.ery> is a finitely
generated sJ, -Module.
(4) ,§<" is locally free if. for each .v G A', there exist an open neighborhood V of
.v and a positive integer 11 such that !Fr — (.»/()"■
An /</«// sheaf of ,af is a subsheaf ,T of .a^-modules of ,af. i.e.. the inclusion
Jf <^-> $£ is a homomorphism of.fi/-Modules. Then J^iU) is an ideal of ,fi/((/) for
every open set U of A-.

u
I I'RS.LIMINARHS
Lkmma 2.11. (I) stf is a finitely generated, cjuasicoherent s£-Module.
(2) If :T and & are finitely generated sf-Modules: both a direct sum !?~ •■[-■% and
a tensor product ,9" ;.<-v-W are finitely generated. If .^ is a suh-.S^-Module of a finitely
generated .W-Module !F. then ^Fj^ is finitely generated.
(3) If ,9~ is finitely generated, then Supp,y~ :=- {.v £ A':5\ * (0)} is a closed
subset of X.
Prook (1) It is apparent from the definition that sf is a finitely generated,
quasicoherent ^/-Module.
(2) For each .v e_ X. there exist an open neighborhood U of .v. elements .v,
(1 < /' < /;) of JiV) and tf (1 < j < m) of &(U) such that 5\ = £" ,..<(*,•),.
and .§;,. - £'," i -'A-(tj)} for each y e U. Then
n ti'
; 1 / 1
and
(^«vn = 5353^.-((-0. 0(/'M
/ 1 / 1
for each y e 0'. Hence, both S^ <;: ,¾5 and f?' ^½.?' are finitely generated.
(3) For .v ^ Supp^. there exist an open neighborhood U of a and elements
Si (1 < i < n) of 9~(U] such that •?",. = ^" 1-^1-(-^)1- for each v 6 6'. Since
,'i*v = 0 by the hypothesis, replacing U by a smaller one if necessary, we may assume
that (.v,-),. — 0 (for each y £ U. !</<«)■ Then ^-0 for each v e t/ and
(7 C A' - SuppiF. Namely. X — Supp.7" is an open set. H
From now on we shall mainly treat quasicoherent and coherent Modules on a
ringed space and. for this purpose, state basic properties of coherent Modules.
Thi-.ori-.m 2.12. Let {X..<if} he a ringed space, and let
Q-^^-U^^^ — O
he an exact sequence of si'->nodules. If two of ft . 'S. and 3? are coherent the third is
coherent as well.
Prook.
Ca.sf 1. .9 and $P ark cohkrknt. Since .*? is finitely generated, there exist an
open neighborhood U of v and a surjective morphism 11: {st'rY - ■ ?h .for each
.v e X. Since 3P is coherent. Kert£ - u) is a finitely generated ^(-Module. Hence,
by replacing U by a smaller open neighborhood of v if necessary, we obtain a
commutative diagram
(.5/( )■•' —^ (V, v^^jf,_—> 0
; . j" I
0 ^.9~l —^.¾ —'i-^S?i. ^0
such that each row is an exact sequence. Here the morphism tr: [sff)'' -^ ,9~i induced
by u is a surjective morphism. (Verify this by considering slalks.) Hence. !? is finitely

2. THHORY 01 SHHAVHS AND C'()H()M()L.()(ill.S
is
generated. Meanwhile, since 5" is a sub-j/-Module of SF. the second condition of
coherency for !F follows from that for S.
Cash 2. 3~ and S ark cohhrhnt. Since S is finitely generated. JF is also finitely
generated by Lemma 2.11. We shall show that for any homomorphism u: {stfi )" —^
%?i . Kerw is a finitely generated .sfL -Module. In order to show that Ker» is finitely
generated, we may consider a small open neighborhood of each point of U. Hence.
we may assume that u decomposes as
Since & is finitely generated, we may further assume that there exists a surjective
homomorphism c: {str)'" -► ^i ■ Now consider the following commutative diagram
0
A
0 ^srL '—^SL^C— ^0
A A X^^ |
' \ I"
0 ^(^f.)'»^=^(j/r)'»,"^=^(,af[-)" ^0
A ' A
«' kir
where h.k.r.s are homomorphisms associated with the direct sum decomposition
(s/L )'"'" = (j/r)"' <h< {stfi )"■ Namely, we have r • h = 1. k ■ s = 1. k ■ h = 0.
;• • .v = 0. and h • r + s • k = 1. Let t = f • c • r + r • k. Since S is coherent,
there exists a surjective homomorphism w: {s^L)r —> Kerr with U replaced by a
smaller one if necessary. Then it is easy to show by the diagram chasing that the
right column {ssfL)p ^> {afi-)" -^^- is an exact sequence.
Cash 3. !? and ?? arh cohhrhnt. First of all. we shall show that S is finitely
generated. Since !F and %? are finitely generated, for each .v e X there exist
surjective homomorphisms e: (j/j-)'" —» &i and u: {$$i■)" —> ^L on a suitable open
neighborhood U of .v. We may assume that u decomposes as (sfL )" -USi- -^ %*L .
Employing the commutative diagram in the case (2). we let t = / • c ■ r + v ■ k.
Then t: {safL )'"'" —> Sv is a surjective homomorphism. Hence. "& is finitely
generated. Next we shall verify the second condition of coherency for S. Let U be
anew an arbitrary open set. and let u: {sfL)" —» Si be an arbitrary
homomorphism. In order to show that Ker» is finitely generated, fix a point v of U. Since
Ker# • u is finitely generated with g • u: {sfc)" —► ^i ■ replacing U by a smaller
open neighborhood of .v if necessary, we may assume that there exists a surjective
homomorphism r: {stfL-)"' —> Ker# •;/. Then there exists a homomorphism of ,sfi -
Modules w: (jtfr)"' —> ^- such that /' • w = u ■ v. Since Sf is coherent, replacing U
by a smaller open neighborhood of v if necessary, we may assume that there exists
a surjective homomorphism p: {stfl )' —> Keru:. Thereby we obtain the following

*(S
I. PRELIMINARIES
commutative diagram.
o *- &i —•*■ &L—?-*. srL *- o
j„
A
It is then easy to show by chasing this diagram that the middle column (safL )1 —i
(.«V)" -^-&i- is an exact sequence.
n
Theorem 2.13. For a ringed space (X.srf) we have:
(1) For a homomorphism u: 5F —> 2? of coherent si'-Modules. Kerw. Coker w. and
\mu are coherent srf-Modules}
(2) For coherent s/-Modules & and &. SF ¢¢.^ 2? and ??om^(!F.c§) are coherent
Si'-Modules.
(3) For s?-Modules & and 2?. we have Sf^^i^.%), = Hom^ (S\-. g\) (/w
each -v £ A') provided & is coherent.
Proof. (1) Since & is finitely generated, so is the quotient Module Imu.
Furthermore, since Im u is a sub-j/-Module of "S. the second condition of coherency
holds for Imw. Hence. Imu is coherent. Since Kerw and Coker » appear in the next
exact sequences
0 —> Ker u —> EF —> Im it —> 0.
O^ImM^f^ Coker u -► 0.
Ken/ and Coker w are coherent in view of Theorem 2.12.
(2) By Lemma 2.11. ^" &v ^ is finitely generated. Since & is coherent, for each
a el we find an open neighborhood U of x and an exact sequence
(*) (j/Lr ^ W ^ ^r ^ 0.
Taking tensor products of §V with the terms of the exact sequence (*). we obtain
the next exact sequence
(**) {W ^ (&c)" ^ &L <**, &l -+ 0.
We remark here the following simple fact. Namely, if & is an j/-Module and { £/;.: k e
A} is an open covering of X. then & is coherent if and only if &~r.-. is a coherent
j/( , -Module for every A 6 A. By the result (1) above, the exact sequence (**) shows
that {.W (*v &)L = 9~f C*V, ^r is a coherent stfL -Module. So !? ($,& & is a coherent
j/-Module by the above remark.
With a natural injectivc homomorphism /: Ker» —> !f. we define lnw/ as Coker/.

2. 1HI.ORY Oh Sill AVI S AND COIIOMOLOOU S
37
The exact sequence (*) also gives rise to an exact sequence
o -► jr**..y(^\sf)i- ^ (¾ )" ^ (%if.
where ^^{.T.&)L- = ^,msjil {.'Fl.'§v) and %r„m.^ {{sJL )".&c) = (.§c)" ■ By the
result (1) above. %?„*,..<*{^.3)^ is a coherent .a/; -Module. Hence. ^flm^{£F .CS) is
coherent.
(3) The exact sequence (*) in (2) gives rise to an exact sequence of stalks
{si,)'" ^> (si,)" %^,^Q.
This implies that the two rows are exact sequences in the following commutative
diagram
0 ^ Honv, {,y,. "S,) ^ Homrf,((rf,)". rS,) —^^ Homrfz((^)'". %,)
A A A
o ^j?w^ ji&i-.&i), >^msJi, {(si„)".&r), ^sromsil {(sic)'".%v),.
Thus, we obtain J?W(.ST 5?), = Horn^ (S\. S\). D
Now we shall discuss cohomologies of ^/-Modules on a ringed space (X.j/).
Let R be a commutative ring. An ^-module I is called an injective R-module if it
satisfies the condition: Given an injective homomorphism of ^-modules /: M —> N
and an arbitrary ^-homomorphism g: M —> I. there exists an ^-homomorphism
h : N —> I such that g = h • f. As a dual notion, we have a notion of projective R-
mochde. Namely, an ^-module P is a projective ^-module if P satisfies the condition:
Given a surjective homomorphism of ^-modules / : /V —> M and an arbitrary R-
homomorphism g: P -^> M. there exists an ^-homomorphism h: P —> N such that
g = f • h. An arbitrary R-module M is isomorphic to an R-submodide of an injective R-
module (cf. [1]). Similarly. M is isomorphic to an ^-quotient module of a projective
.R-module.
These notions of ^-modules can be generalized to the corresponding notions of
^/-Modules. An ^/-Module Jf is called an injective si'-Module if. for an arbitrary
injective homomorphism of ^-Modules <$'■ ^-^ and an arbitrary homomorphism
f.y^.y. there exists an ^/-homomorphism Q-.'S-^Jf such that i// = 0 • <p. As
a dual notion, we can define a projective si-Module.
Lkmma 2.14. Any si-module 5F is embedded into an injective si-Module as a sub-
si-Module.
Proof. Any stalk .9~, of 3~ can be embedded into an injective si, -module I, as we
remarked above. For an open set U of X. let <f{U) = Yl,e_t ^- wmch is an sf(U)-
module by the rule: u£ — (a,c,-),ei for u G si{U) and £, = (£J,6r € J^{U). So
{.y{U); U C X} is a presheaf .y of ^/-modules. It is easy to show that ,y is. in fact,
an ^/-Module. We shall show that .y is injective. that is to say. u*: Homrf(T. Jf) —»
Homrf(,?. Jf). if h-+ if • u. is surjective for any injective homomorphism of ^/-Modules
u: 3? -> ST. Note here that rlom^^./) = [Lev Hom< ^,.J,).,y, = I, for
each .v € X. This can be shown as follows. Let X(/ be the underlying set of X
with the discrete topology, and let f: Xcl —> X be the identity mapping as sets.
Then /' is continuous. Let si' be a sheaf of commutative rings on X,i defined by
U h-> si'(U) = PJ t.si,. Then (X,/.^') is a ringed space and f is naturally

18
I. 1'RFLIMINARIFS
a morphism of ringed spaces. The associated homomorphism of sheaves of rings
sf -> j\9?' is given by a e stf{U) «-> (a,)^v € $?'{U). If J" is an ^'-Module on
X,i denned by J"{U) = T],et A- then ,J? = /^. By Lemma 2.9. we have
Honv(2?. JO ^ Hom.rf.(/*,?.y) = fj Hom,A (3^.^-).
where we note that f*&(U) = iLef^- Similarly. HonwpT.JO =
ri\fcA- Hom.^ (^"v..yv). Since «,: 5^ —> J?\ is an injective homomorphism of
,s^-modules, u*: Hom^^-./v) —> Hom,^ (S\. ,J^J is surjective. So ;/* =
rivgA'"*- Horn.,/!/./) —> \ioms^(^.,_y) is surjective as well. Hence. J^ is an
injective ^/-Module. Meanwhile, define 3~{U) —> live; ^ by .v h-> (,vv),er and
rixef^ ~* fixe; A- as tne product of embeddings ,9\ ^-> /v. Then the composite
of these homomorphisms .9~(U) —> ^(U) is an injective homomorphism. So J*" is
embedded into an injective ^/-Module ,y. D
Let ^" be an .af-Module. An injective resolution of & is an exact sequence
such that ,f (/ > 0) is an injective .s^-Module. Given a homomorphism of si-
Modules a : S*" —> S*"' and an injective resolution of 5"
0 _+ ^' 1, ^'0 ^ jr/l ^ ^/2 _ . . . A^' jr/« ^ . . . .
there exists a homomorphism of ^/-Modules a": J?" —> J8-'" (« > 0) such that
n' ■ a = a{} ■ n and A'" • a" = a"1' • A" (« > 0). In fact, since J2"'0 is injective.
we find a" with //' • a = a" • >/. Suppose a' (0 < i < n) is constructed. Look at
A'" • a". Since A'" ■ a" ■ A"-' = A"' • A'"-' • a"' ' = 0. we have a decomposition
A" • a": J" — Im A" -> j""''. Since Im A" is a subsheaf of J"''' and Jf'"'^ is
injective. we find a"' ': J"''' -> J""1' with A'" • a" = a"T' • A". We call {a"}„>(,
an extension of a. Let {a'"}„>o be another extension of a. Then there are s4-
homomorphisms {/?" : Jn{' -> /'"}„>„ such that q° - a'0 = /?° • A° and a" - a'" =
A'»-i .^« -i +[}».A" («> 1). In fact, since (a()-a'0) •// = //'-(a-a) =0. there is a
decomposition q° - a'0: J^° —> Im A0 —> J^'°. Hence, there exists a homomorphism
/i°: Jx -^ ,J'{) such that q° - a'0 = //° • A0. Suppose (i< (0 < i < n) is constructed.
Then we have
(a"-a'"- A'"-1 -/r ')• A""1 = A'"'1 .(a""1 -a'"' ' -0"-' -A" ')
= A'"-' ■ (A'"-1 ■ pu~2) = 0.
Hence there exists a homomorphism /j": J^"'1' —> J"""' such that a" - a'" - A'"-1 •
j}n-\ _ ^«. ^« ^ye tjlen say tnat tne extensions {q"}„>0 and {a"'}„>o are homotopie
via {(]": S"u -^ ,y"<}„>»).
Given an ^/-Module ^. there is an embedding tj: ST —> J^° of ^" into an
injective ^/-Module J^° by Lemma 2.14. Considering Coker;/ for &. we see there
is an embedding Coker^ —> J^1 into an injective i/-Module J^'. By composing this
embedding with the natural surjection ,y{) —> Coker^. we have a homomorphism
A0: Jf{) -^ .yK. Next consider CokcrA0. Repeating this argument, we obtain an
injective resolution of &~.

2. THF.ORY OF SHF.AVF.S AND COHOMOLOG1F.S
V)
We can find injective resolutions in more restricted situations
Lkmma 2.15. Let
0 — W\ 4 9r2 L gry _^ o
be an exact sequence of sf-Modules. Then there exists an injective resolution of 9~-,
(/-1.2.3)
1,, n A" , A' A""1 „ A"
such that the following diagram is commutative:
0
I
1"
il<
&y
I
0
'/i
'/:
'/"-
0
I
if
JS°
I
0
A','
A':
a"
0
I
I"1
^,1 - •
I/'1
^' - •
I
0
0
I
u 1
1«
I/'
—J- *s ^
I
0
mVzcw cw7? column is an exact sequence and accordingly. .J?" = Jf" ^ J^" /«r «// n > 0.
Proof. Let /71: 9~\ -^ J\ be an embedding into an injective ^-Module. Consider
a homomorphism (a.-/71): 9~\ -5^^^,0. Let'S = Coker(a. -/71)- Considering an
embedding & ^ Jf into an injective ^/-Module and composing this embedding with
the natural homomorphism Jf" — ,9¾ (!< J^° — ^ (or 5¾ —> 5*2 'I' ^° —>• ■?"■ resp.). we
have a homomorphism a'°: J^° — J^ (or n1-,: SFi—> Jf. resp.). where J^° —> Ss * ^°
(or ,9; —^^2^-^1°. resp.) is given by a 1—* (0. a) (or b t-> (/).0). resp.). Both a'0 and 77,
are injective. Let J? = Cokera'0 and let /?'°: J^ —> J? be the natural surjection. Since
J^f is injective we have a homomorphism 7: J^ —> J7,0 such that 7 • q'° = 1 (= the
identity morphism of J^0). Hence, there is a decomposition Jf — a'°(.y\}) ¢- Kei"7.
where Ken/ = ,/. Kei'7 is apparently an injective ^/-Module. Here there is a
homomorphism n\ :^^/ such that /?'° • n'2 = n\ • /?. though n'y is not necessarily
injective. So we take an embedding r:^1-*! into an injective j/-Module and put
J{1 = / * Jf. 73 = (n'y v): y-s - y>. J[l = J e 91. m = (rj'2. v • /i): ?2 - J{1
and q° = (q'°. 0): J")" — J^'. Then we have the following commutative diagram.
0 0 0
0 *- &~\ —^ 5^ ——^ &~n ^ 0
'/I '/: '/"
0 Ji -^ f$ -^*s* —-0
Let ,?7 = Coker/7; (./' = 1.2.3). Then the above diagram yields an exact sequence
0-4¾^¾^ 'Sy — 0.
0 ->
0 ->
0 -

40
1. PRELIMINARY'S
Repeating the foregoing argument, we can embed this extict sequence into an exact
sequence of injective ^/-Modules
o -+ ,r\ ^+ ,y\ C .y\ -^ o.
Again repeating this argument, we obtain an exact sequence of injective resolutions
as required in the statement. The «th exact sequence of this injective resolution
o -+ s1; ^+ si ^+ j'i -+ o
splits. Namely, we have J^' = Jf'{ ^J'^. □
For a sheaf .y on X and an open set U. we often write r(U. 9~) instead of SF(V)
and call it the set of sections of S*" over U.
For an exact sequence of j/-Modules
0 -+ .¥ -^ rS -^+ 3? -+ 0.
the sequence of Y{U..s/)-modules
o — r(t/.so f^: r(u.%)k(Vy{v.ssr)
is exact. (In fact. ,1 is isomorphic to Ker#.) Taking an injective resolution of an
^-Module &
and applying the functor r(t/. •). we obtain a sequence
o-+ nt/./jAny./0) '^r{u.,y]) ^+--
,y^ r{u.sn) ^ •" .
where e = /?([/) and 5' = A'{V) (i > 0). Here 5' ■di-[ = 0 (/' > 1). Thereby we
obtain a complex
(*) o^r(c/.^,,)^r(c/.^1)^ ...,5l+' r(c/..r")^-" .
Since S' ■ S''""' = 0 we have Im^'~' C Ker<5'' (/ > 0). where we put S~] = 0. We
denote the r{U.stf)-module KerrJ'/Imr)'-1 by H'iU.F). H'(U.,9") is independent
of the choice of an injective resolution of 9~ by the reason we explain below. Take
another injective resolution of,?".
in
in
As we have seen earlier, there exist ^-homomorphisms a": ,/" -+ J?'" and /?": J?
J" (H>0)suchthat/?' = a,V./?=^V-A'"-a" = a"+l -A", and A" •/?" = /?"■' -A'
Furthermore, there exist .s/-homomorphisms p" : J"'1' —> J^" and a": ,y,n'' —> J*""'
(;j > 0) such that 1 - /?° • q° = /;° • A". 1 - a" • /?" = a0 ■ A'°. 1 - //" • a" =
A"-' • />""' + /)" • A", and 1 - a" • /?" = A"'"1 • a"-] + a" ■ A" (« > 1). Let

2. THKORYOl Sill, AVIS AND COHOMOLOGIl S
41
e' = rj'{U). 5" = A"([/). a" = a"(U). b" = fi"{U). r" = p"{U). and ,v" = o"{U).
Then between the complex (*) and a complex
(**) o^r{u.,y°)Cr(u.,y,]) 4 ■ ■■"'-V r{u.,y'")C---
there are the following relations:
- e' = a°-e.e = b° ■ e'.
6'" ■ a" = a"'1 -S".S" ■ b" =/>"+' • <V" {n > 0).
(* * *) I 1 - h° ■ a° = /•" • 8°. 1 - «" • b° = ,v° • S'°.
1 -b" -a" =S"-] ■)•"-' +r" -S".
, 1 _ a" ■ b" = S'"-] ■ x"~] + s" ■ (V" (n > 1).
First of all. since Ker<5" = Ime and Keiv5'° = Ime'. one gets Ker<5" = Ker<5'°
via homomorphisms «" and />°. from which it follows that Hl)(U.&~) is independent
of the choice of an injective resolution of &~. The homomorphisms a" and b"
induce homomorphisms Ker^'/Im^"-1 —» Ker<5'"/ Im<-5",_1 and Ker^"7Im<5"'~' —>
KerS"/ Im^"~' respectively, and it is easily shown by the relations (**) that they are
the converses of each other. Hence. H"{U.!F) (n > 1) is independent of the choice
of an injective resolution of ^. We call H"{U.Sf) the «th cohomology of &~ over
U. Below, we mainly treat the case U = X.
Let f: 9~ ~+ .?" be a homomorphism of ^/-Modules. Take injective resolutions
of & and %.
o ^ ? 1+ ,y» 4 ^ 4 ... AV s» ^ ... .
Then there exists an extension {/": .f" ->/"}„>oof f such that y" ■ f" =/"tI -A".
This extension induces a homomorphism of r(t/.j/)-modules H"(U.f): H"(U.&')
~+ H"{U.&). As we have observed earlier, two extensions {/"}„>(> and {,/'"'}«>()
of / are homotopic to each other. Thereby, we can readily show that H"(U. f) is
independent of the choice of extensions of / . If /: EF ~+ 'S and g : "§ ~+ 3? are
homomorphisms of ^-Modules, then we have H"{U.g- f) = H"{U.g) ■ H"{U. f).
Next we shall consider a morphism of ringed spaces cD = {f.ip): {X.stf) —>
(Y.38). To simplify the notations, we let f represent the pair {f.ip). For an exact
sequence of ^/-Modules
the sequence of ^-Modules consisting of direct images by /.
o -> j\f '4* /„3? '41 j\%f
is an exact sequence. (The functor /'* is. therefore, a left-exact functor.) This is the
case because
is an exact sequence for an open set V of Y.

42 I. PRELIMINARILY
Take again an injective resolution of &.
o^,t^ ,y() ^ ,y' £ ■ ■ ■ *V ,y £ ■ ■ ■
and consider a complex of ^-Modules
A ^-Module Ker/^A"/Im/*A"_1 (n > 0) is independent of the choice of an
injective resolution of & up to isomorphisms. This can be shown by the same
argument as for H"{U.^). We denote the ^-Module Ker/*A"/ Im/*A"~l by
R" j'*!? and call it the »th higher direct image of & by /'. We have ^°/'*5r = j\9~.
For a homomorphism a: &~ -+ rS of j/-Modules. we can define a homomorphism
R" ft(a): R" f\.T -+ R" f,S'. As in the previous case, we have R"j\{fi • a) =
R"\f\{Si)-R"'fAa).
Theorem 2.16. Let {X .$?) be a ringed space, and let
0 -► &x -+ ,9s ^+ ^ -> 0
be an exact sequence of si-Modules. Then we have:
(1) For each n > 0 there is a homomorphism of"T{L'. s4)-modules S": WfU.^-f)
-+ H"] X{U.,7'\) such that the following sequence is an exact sequence:
0^H°(U.&i) ^Ha{U.&-2) ^Hl\U.y?) '^+ H]{U.Fl)
'±H\U.?i) ^+ Hl{U.&3) ^+ > H"{U.9-X)
'-^H"(U.FZ) '-^ H"{U.^)'t H',r\U.^x) -+■■■.
where U is cm open set ofX, a" = H"{U.a). and (J" = H"{U.fi).
(2) Let /': (X.sf) -+ {Y.3S) be a morphism of ringed spaces. Then there is
a homomorphism of 2'-Modules 3": R"/*^ -+ R"+{ f\9~\ such that the following
sequence of 3B-Modules is an exact sequence:
0 -> /?°/*^i ^ R°f\F2 ^ R°f*9~} ^ R\f*?\
^ R]f\^2 ^ R]f,F* ^ > R"f\^\
^ R"f\&i ^ R"f*?i ^ R'H xf*&\ -*•■■■
where we denote R" f '*(a). /?"/*(/?) by a"./?" respectively by abuse of notation.
Proof. (1) We use the injective resolution of an exact sequence which we
constructed in Lemma 2.15. Let V = V(U.S[). M' = Y{U.J[). and N' = r(U.J"{).
Moreover, in order to avoid complexity of notation, we write a'./?' for r(U.a').
r( £/./?''). respectively, and d' for r(t/. A';) (i > 0. 1 < / < 3) indistinctively. The
following sequence is a complex
L':0-^La^L] ^1-. • Cl" ^---.
Similarly, we have complexes M* and N*. and a' = {a": L" —» M"}„>0 and

2. THEORY OF SHHAVHS AND (OllOMOLOOIF.S
43
P' = {/i": M" —> N"}„>{) are homomorphisms of complexes of R (= T{U.stf))-
modules. Namely, we have d" ■ a" = a"" • d" and d" ■ [!" = /i"+l • d" for each
n > 0. Furthermore.
0 -> L" ^ M" ^ A"' -^ 0 (« > 0)
is an exact sequence. In fact, this follows from the splitting .J?" = .Jf" tB J^'. In sum.
we have an exact sequence of complexes
0 _> L' C M' C Nm -» 0.
Let H"{L*) = KerJ'VImJ"-'. where </"' = 0. Likewise, we define H"{M*) and
H"{N*). Suppose r e A™ with d"{z) = 0. Then /?"(>■) = r for >• G M". Since
/?"+l • d"(y) = d" ■ P"{y) = d": = 0. d"{y) = a"+1(.v) for some .v e L"+l. Since
a"+2 • (/'"'(.v) = d"+t .q"tI(.v) = c/"H •</"(>') = 0 and a"+1 is injective. we have
x e KerJ" '. Then the residue class [a] of .v in H"U(L') is independent of the
choice of y and determined only by z. Furthermore, [x] is independent of the choice
of a representant z in the residue class [z]. If we define a mapping S": H"{N') —>
//"+I(L*) by <5"([_]) = [a], then S" is a homomorphism of ^-modules. It is an
elementary diagram chase to ascertain that the following sequence of ^-modules and
^-homomorphisms is an exact sequence
0 -> H{\L') ^ H\M') C H°{N') ^ H]{L') -> • ■ •
->H"(L*) '-^H"(M') ^ H"(N')^ //"+I(L*) -> • • ■
The details are left to readers. This exact sequence is nothing but the cohomology
exact sequence stated in (1) above.
(2) A similar argument to the one above can be applied to prove that the sequence
of ^-Modules consisting of higher direct images is an exact sequence. A crucial point
is that since JF" is an injective Module we have a splitting J^-," = J*"," <$, J^". □
If J*" is an injective ^/-Module, then
o-> j^ ^,y->0^ •••
is an injective resolution of J*". Hence. H"(U. J*) = (0) for each n > 0. Here we shall
review again the cohomology groups from a bit more general viewpoint. Let (X. stf)
be a ringed space, and let U be an open set of X. A family of functors {K"(U. •)}«><)
which assign a T( U. ts/)-module K" (U. SF) to an ^/-Module &~ is called a cohomology
functor or a cohomology theory if {K"(U. -)}»>o satisfy the following three conditions:
(1) To a homomorphism of ^-Modules/': ,9~ —»3? there corresponds a r{U.srf)-
homomorphism K"(U.f): K"{U.^) -> K"(U.&). and K"{U.g ■ f) = K"{U.g) ■
K"{U.f) and K"(U.id9) = idA-„(r^).
(2) K^iU.S1") = &{U) and K"{U.,f) = (0) (all n > 0) for every injective
^-Module J.

44
I. PRELIMINARIES
(3) For an exact sequence of ^/-Modules
0 —► .91 -^- .9~2 -^- .^ —>• 0.
there is a homomorphism of r(t/.^)-modules 5": K"{U.Fy) ~+ K"]](U.^X) for
each n > 0 which makes the following sequence exact
0 -> A:"(t/.^i) ^ K°{U.^2) t K0(U.&3) ^K\U.F\)
'-^ K\U.^2) 't > K"{U.^i) '-^ K"{V.&2)
Ck"{U.^)'^ K"fl (U.,^,)^--- .
where a" = K"{U.a) and fl" = K"{U.(1).
Under these circumstances, we have the following result.
Lemma 2.17. Let { K" (U. •)}»>() be a cohomology functor on a ringed space (X.sf).
Then the functor coincides with the cohomology functor {H"(U. -)}»>o defined in
Theorem 2.16.
Proof. Let & be an ^-Module and let
0 -* & ^ S° ^ ^1 ^ ■ • • -' J" ^ -J"^ -+■■■
be an injective resolution of,!?". Setting 3V" = ImA". we have exact sequences
o -> & -^ ,y(> -+ x" -> o.
o-^x"-1 -> J^" -^ JT" -^0 (« > 1).
Noting that H'iU.J"') = K'{U.J^") = (0) (for each / > 0 and each h > 0). we
employ the condition (3) to obtain the isomorphisms
HX{U.F) ^ Cokerp^t/) ^^°(C/)) = ^'(f/.^").
H"n(U.9r) ^ H"{U.JT°) = H"-\U.Xl) ^---^ Hl{U.3T"-]).
K"'\U..9~) = K"{U.X(>) ^ /^"-'(t/.X1) ^ ... ^ ^'([/..X"-') (for every «>1).
Hence, we have an isomorphism <^"(Sr): H"{U.?F) ^> K"(U.&~). By the
construction of an isomorphism tp](&'). we can easily verify >?"{&) ■ H"{U. f) = K"(U. f) ■
<p"{y) (for every n > 0) for a homomorphism of j/-Modules /: ^" —>.?". D
Since the cohomology groups H"{U.SF) are defined in terms of an injective
resolution of &~. it is not easy to compute H" (U. ^) for specific examples by pursuing
the definitions. For this reason, we define Cech cohomologies and make
supplementary use of them in concrete computations. In the subsequent arguments, we
consider the case U = X: the case U c X can be treated in the same fashion.
Let it = {l/,},6/ be an open covering of X. For .v = (/0 i„) e /"+1. we set
L\ = Uk n [/„ n • • • n £/,,,. An element a = (a(.v)\e/„.i of ILe/*-' ^{Vs) is called
a Cech n-cochain with coefficients in ,9*" if it satisfies the following two conditions:
(i) For .v = (/().(1 /„). q(.v) = 0 provided tA = 0 or /, = u for some
j.k:jfk.
(ii) Excluding the case (i) above, we set a(.v) = (<„(())■'It(I) ^n(n)) f"or any
permutation a of {0. 1 «}. Then a{a(s)) = sgn(a) • a(.v).

2. TH1-.ORY OF SHhAVI.S AND ("OHOMOLOGll S
45
The set of all Cech «-cochains with coefficients in 9 forms a T{X. j/)-module
C"(U. y). For an ^/-homomorphism f:9-^>^. we define a Y\X. ^-homomorphism
/": C"{iX.9) ~+ C"{iX.&) by
(/''a)(.v)=/(t/,)(a(.v)).
We also define a T{X. si)-homomorphisin d": C" (U. 9) -> C"4' (U. SH by
(rf"a)(0 = X;(-l)'a('/)U,
/-o
for q e C'^ILSH and t = (/0. • • ■ . /„ + i). where 11 denotes (/0. • • • . ir ■ ■ ■ . in+\) which
is obtained from t by removing /;. It is a straightforward computation to show that
d"^1 • d" = 0 for each n > 0. Hence, we have a complex with boundary operators
C(){U. 9) ^ C' (U. 9) ^ • • ■ C C" (U. ^)
^C"M(U.^") — •••
which we denote by C'{!d.9). The cohomology group of this complex is //"(11. 9).
Namely. H"(ii.9) = Ker<-/"/Imd"~]. We denote Kera"'. Imf/""1 by Z"(U.SH-
i?"(U. .9"). respectively, and call elements of respective groups Cech n-cocydes and
Cech n-cohoundciries. In particular. //°(U.9) = T{X.9). (Verify this.) An sf-
homomorphism /': 9 —> S/ induces a homomorphism of Cech complexes
/" = {/"}«>(>: C-(U.^)->C'(U.5?)
and accordingly a r(X.^)-homomorphism /": H"{ii.9) -> H"{iX.&) (« > 0).
Let 23 = {V,}/€l\ be another open covering of X. We say that 23 is a refinement
of it (notation: it < 23) if there is a mapping ip: A —> / such that K; C C/^(/) (for
each A e A). Utilizing this mapping tp. we define a homomorphism of complexes
¥>' = {<p"}»>a: C'(iX.9)^C'(V.9)
in the following fashion: For a e C"(U.^") and t = (A(). ■ • ■ .1,,)- (y>"a)(0 =
a{<p{t))\r,. where <p(/) = (^(A0) ^(A,,)) and K, = VA[1 n • ■ • n KA„(C [/^,,). If
i//: A —» / is another mapping with V, C [/v/(/) for each ieA. then y/' = {y/"}„>o
is homotopic to <p*. In fact, if we define/;": C"n(ii.9) -+ C"{<8.9) by
II
(p"a)(t) = ^(-l)'Q(¥>Uo). ■ ■ ■ . ip{k,). y/(A;). ■ ■ ■ . ^/(/-,,))1,-, .
/-()
' = Uo A„)
and /.>-' = 0. then we have a relation
y," - ^» = J"-' . /,"-' + p" . d" (n > 0).
(Verify this.) Hence, two homomorphisms H"(iX.9) —> //"(23.^) induced by y>"
and y/" coincide with each other. We denote this homomorphism by a<uu. For
refinements of open coverings it < 23 < 233. we have (7¾¾¾ • a>mi = ^aim- So we can
define an inductive limit of {//"(it.9).owj) and write H"{X.9) = lim H"{!d.9)

46
[. I'lUiLIMINARILS
l.for each n > 0). We call this group the nth Ceeh cohomology. Here we have
/?"(.V..T) - Hn{H.T) - T[X.&).
Wc have thus defined the ordinary cohomology group H"(X.^) and the Cech
cohomology group H"{X.:9). So it is now our task to examine conditions under
which these two cohomologies coincide with each other.
Let it be an open covering of X. We define a presheaf f "(il.JF) by assigning
C"(V nit. '9~\i•} to an open set V of X. where V "lUis an open covering {Vn U,}iel
of V and C"(V P M..W\\ ) is a Y(V. „o/)-module consisting of all Cech n-cochains
with coefficients in ?#\\ . It is easy to verify that %"{i\...9~) is an ,a/-Module. The
boundary operators
d"(V): C"{V nit.^j, ) -* c" + i(v nu.,9^1,)
give rise to a homomorphism of ^/-Modules d": &"(iA.,9~) —> %"'' '{il. J?"). Hence.
we obtain a complex %'*{U. 5») of js'-Modules
0 —r"(U.y) ^'(il.sn ^ ■-■"—' f"(11..5^) ^---.
Here if we define a homomorphism of ^-Modules j: .9 —> W[){U.. ,9~) by a e ,9' {V) <-*
(«|rp,r, )k_/ £ C°( I'nii.^[i ). we have the following result,
LhMMA 2.18. (.1) Complementing the complex %'*iii..9~). we obtain an exact
sequence, which we call the Cech resolution of\9~
0->Sr-'>&°{U.9-) "^WUil.F) llX ...
(*)
"-» %"{<&.,t) ^r^'(u..y") ->■■-.
The complex C*{W.9~) is obtained from W{11. .9") by applying the functor T(X. •) to
each term and homomorphism of&*{W.EF).
(2) For an inject ire resolution of' 9r
o ^-/'^^ ^..,^>v<A;....
there exists a homomorphism 0": %"'{il.&') ~* J" in > 0) such that f/° ■ j = n and
0"'' • d" = A" ■ 0" for each n > 0. Hence, there exists a V(X. s4)-homomovphism of
cohomology modules 0'^: H"{ii..9~) -^ H"{X..9") for all n > 0.
Proof. (1) For .V t X and n > 0. an element a of®'" (11. .9), is represented by an
element a ot'W'iil.,9~)(U). where U is an open neighborhood of .v. We may choose
U so that U c £/(i for some Oe/. Then U D U; = V n £A for .v == (/(, /„_i) and
.^ - (0. A, /„ i). Define an element /i of £""-l(U..^)((/) by /i(^ J - <*(.?}. Then
we have
/i «
'> 0 i -0
where ? - (/() /„). tk = U0 t\ /,,). f = (0. /o /„). and t,- = (0. /{l //,.
.... /„). Suppose d"(a) = 0. We may assume d"{a) = 0 by replacing V by a smaller

2. 111101« ()1 SHI. AV1-.S AND COI lOMOLOOIhS
47
one if necessary. Then
n
(,/<*)(/) -- a(,)- J2(~ 1)M'A ) = 0.
k i)
Hence, a - d" fi. Thereby. Kerr/" = \md"~] for each n > 0. By a similar
argument, we can show that Kerr/" ^ ,9~'. The last half of the assertion (1) can be
readily verified by the definition.
(2) Extending the identity morphism id> : ,9~ ^+ .9~. we obtain a homomorphism
0": %"' {tt..9~) ->,y («>0). Two extensions %"'{U. &) -> ,y" of id ? are homotopic
to each other. This can be shown by mimicking the argument which we applied to
two injective resolutions of &. □
The homomorphism 0[\: //"(U.//") —> H"{X..9r) which we defined in Lemma
2.18 satisfies ()^ ■ <t>jj1( = 0[\ for a refinement of open coverings U < 2J. So we define a
homomorphism of r(^.,^)-modules 0": H"{X.W) -+ H"(X.&) by 0" = lim «(',.
A sheaf c? on X is called a scattered sheaf (or a fasquc sheaf) if the restriction
morphism ;>lx\ r(X.S) —» r{U.<S) is surjective for any open set U of X.
Lemma 2.19. (1) An injective sJ-Module is a scattered sheaf.
(2) For an exact sequence of si-Modules
o -+ $~\ ^+.% ^+ 9\ -+ o
with a scattered sheaf fF\. the sequence
0 — riU..^)'^ V{U..¥2)II[V Y(U.fF^) -0
/,v exact for anv open set U.
(3) Let
0 -+ .¥{ ^+ ,<?2 ^+ ,5^ -+ 0
he an exact sequence of si - Modules. If ,9'\ and 5*2 are scattered sheaves, then so is ,9~}.
Proof. (1) Let A be the sheafifying space of si and let n: si -+ X be the natural
surjective morphism. For an open set U of X and F = X — V. define a presheaf of si -
modules .8/(/.] as follows: For an open set V. si\V\( V) = {s : F n V —» n~ ' (F n K): .v is
a continuous mapping and a section of n\. where F C\V and n"] [F (1 V) are endowed
with the induced topologies of X and si. respectively. It is easy to show that si\i-\ is
an j/-Module and (.^/])x = 0 for each .v e U. Furthermore, a homomorphism of
.^-Modules/;: si -+ siy.] given by p(V): s e si( ]') v-+ s\fr] e .£/[/■]( r) is surjective.
(Show that the homomorphism of stalks pK: six -+ (siyF^)x is surjective.) Denote
by sin ] the kernel of/;. By the definition. si[r](V) = {s e ^(I7):^/.,--,! = 0} and
(.8/(( ])x = 0 (for each .v G F). We obtain, therefore, an exact sequence of .(/-Modules
0 -+ .s/[(, -+ sJ A j/t,, -^ 0.

4S
I. PRELIMINARIES
For an 5/-Module 9~. we have Homi(/(rf[(].f) = T{U..9'). In fact, since
sf[i-]{U) = sf(U). the mapping 0 which assigns ^(^)(1.^-)) G T{U.9~) to <p g
Homrf^c].^) is an isomorphism, where \^{r) is the identity element of the ring
srf(U). In fact, it is easy to show that 0 is injective. In order to show that 0 is
surjective. we define ip G Honv^rj.-SH for /' G Y{U.?F) in the following way:
y>(F)(.y) = .s7;rnr,t (/)(.v G sf[n{V)) for an open set V of X. If V n F = 0.
then F n U = V and .v/;,n, v{f) G ^(F). If F n F ^ 0. then s\Fnr = 0 and
Supp(.v) := {.v G K:.vv ^ 0} c Fn £/. Hence. .vprnr.r(./ ) is regarded as an element
of 9~{V). Anyway. <p(V)(.s) is well defined as an element of &{V). Apparently.
0(ip) = f. So 0 is an isomorphism. Now let J be an injective J^-lVlodule. Then
we have an exact sequence
0 *- Hom.t/(.»/[/.-]. .y) * Honw(j/. J^) * Honv(.a/[r]. J?) >■ 0
i y
r(A'.j?)———^r(u.s)
This implies that ,y is a scattered sheaf.
(2) Considering^|r instead of ^-,. we may assume that V — X. Since Y{X .•)
is left-exact, it suffices to show that (1{X): 9~j{X) —> 5^(A') is surjective. Fix an
element u G 5^(^). Let 6 be the set of all pairs (V.t) of an open set V of X
and an element t g 5^( F) with fi{V){t) = p\x{11)- Introduce an order in © by
setting (F(.fi) > (('V^) if and only if F, D F? and f2 = fi|i%. Then 6 is an
inductively ordered set. So by Zorn's lemma. 6 has a maximal element, say (U.t).
If U = X. then fi{X){t) = u. Suppose U ^ x. For .v G X - U. there exist an
open neighborhood F of .v and an element t' G 9~i{V) with [l{V){t') = />rx(}<)-
Here ?'|rnr - '|rnc = ^(^ n t/)(.v) with .v e ^i(F n U). Since 5^ is a scattered
sheaf, .v = .vo|rnt' for some .v0 G 9~\{X). Now define t G ^2(^ U U) by fJL' = t and
f|i = f' -a(F)(.v0|r). Since (Fu t/. ?) > (£A f). this contradicts the maximality of
an element {U.t). Therefore. ji{X) is surjective.
(3) This part is straightforward. We omit the proof. □
Corollary 2.20. Let & be a scattered srf-Module. Then for any open covering U
of X. //" (115^) = {Q)for each n > 0. Hence. Hn{X.&) = (0) for each 11 > 0.
Proof. Sheaves ^"(il. 9") which appear in the Cech resolution of & are scattered
sheaves (cf. problem 1.2.4). The result as stated above follow from Lemmas 2.18 and
2.19. D
Consider an ^-Module 5" and its injective resolution
5T JL, J^> ^ J I ^ . . . Al^ jf» ^
Let U be an open covering of X. The Cech resolutions of & and J^' for each j > 0

2. THEORY OF SHI-.AVhS AND COHOMOLOGIHS
with respect to U give rise to the following commutative diagram
0 0 0
Y
A ^jra—*!L
0 ^(11..^)-^^(11.^)-^-
ii"
±^e>im <^o\_a_
0 >-f'(U.^")—-+&l(U.Jf ")■
, I Y
-^ W"(U
/
.J"')-^
Y
r/'"-
</'"-
o—^r"(u.^)
r""(U.^°)
(M A1
</'"-
-^-g"" (U.J"') —
where all rows and columns are exact sequences. Applying the functor Y(X. •) to
this commutative diagram, we obtain the following commutative diagram
0
0
0
o » r(x. 9-) —E-+ r{x. s°) -^ ■ ■ • -^-^ r(x. ,y<) -^-^
0 ^€^{^.9
<Vw QT\ ^C'Hil.J'0) —
0
0
0 ^CI(U.^)^^^CI(U.^°)^L^
,/1
-^.-C"(U. J"')—
c'(ii j™)
0 -■ >C'"(U.
-^C'"(n.Jrl))
-^C'"(U. J™)'
where all columns except the leftmost column are exact sequences by Corollary 2.20.

^0 1. PRFL1M1NARI1.S
Furthermore, the diagram with the top row and the leftmost column replaced by
the zero row and zero column is regarded as a double complex with two boundary
operators. To be precise, a collection of /^-modules and /?-homomorphisms C =-
{€'"".d'": Cm" -y C"""". 6": C'"" -> C'""1: m > O./i > 0} is called a double
complex if d'" ' • </'" = S"' ' • <V = S" • <!'" + d"' ■ 6" = 0. In order to regard the
above diagram with the first row and the first column replaced by the zero row and
the zero column as a double complex, we have only to take [C"" = C'"(il. ,f").
(l„,. Cm„ ^ Cn,-\M_ (_i)»v^„. C'"" -> c"'"'1: m > 0./7 > 0}. Given a double
complex C = {€'"". d'".S"}. we can produce subcomplexes as follows:
(1) For each n > 0. C,*" = {C'"'.d: C" -> C'" '"},,>(). We denote the
cohomology group of this complex by //"'".
(2) For each m > 0. C'/}* = {C"<>'.6: C'"'' -^ C'"'''1 },,>„. We denote the
cohomology group by H'/'f".
(3) C* = {C'.A: C -> C'- '},,>(,. where C = £„,,,, pC'"" and A = d +S.
We denote the cohomology group by Hr{C').
(4) K* = {K'\ d : K'' -> A-'" ' },,>„. where K<' = Ker(f>": C'-l) -^ C '). We denote
the cohomology group by H''(K').
(5) L" = {U'.S: U — £/''},,>„. where Z/ = Ker(r/: C"'' — C1''). We denote
the cohomology group by H''{L').
Lkmma 2.21. Suppose H'/'" = (0) for every pair {nun) with in > 0 and n > 0.
Then the infective homomorphism of complexes L* —> C* induced naturally hy the
double complex C gives rise to an isomorphism H''{L') —> HP{C*) for each p > 0.
Proof. Note that 6 = A on the complex L*. Let TV* = C*/L*. Then we have
an exact sequence of complexes 0 —> L* —» C* —> A/* —* 0 which yields the following
exact sequence (the proof being the same as in Theorem 2.16):
0 — H"{f) -> //"(C) -» //°(A/*) — H\L') — //'(C")
-^ //'(A'*) -> ■•• -^ //''(L*) -^ //''(C*) — //''(A/*)
-> //'"'(//) -^
If H''{N') = (0) for each /) > 0. we have an isomorphism H''{f) -^ HP{C%) for
each p > 0 by the above exact sequence.
Define here a subcomplex N* {h > 0) of TV* by N* = {Nj'rA: A//,' -+ A//;'' },,>„.
where A/[ = ^„>/, A"' "" with N''-""= the image of C'-"-" and A is
the'boundary operator induced by d + S. Apparently. N' = A/* and N^ = (0) (/; < h).
Furthermore. Nf , is a subcomplex of N*. and N*/N* l is a complex as exhibited
by
0 »- • ■ >► o ^ A/°'' —»- A/lJl '' ■> ■ ^ A/" ''■!' s
C()77KerJ C17' C"-''''
Hence. H'^N^/N^ ,) = (0) for each p<h. H''(AZ/'/A/'h) = (0). and H"{N*/N*U)
= H'l~l,h = (0) for each » > /; by the hypothesis. Applying these results to the
cohomology exact sequence which is given by an exact sequence of complexes
O-A/^-A/^A/,;/^, -+0.
we obtain H"{N*) = H"{N*, ,) for each n > 0. each h>0. D

2. i iieory of sheaves and cohomologies
51
Summarizing the above results, we obtain the following result.
Theorem 2.22. Let J he an .tf-Module over a ringed space [X .srf). and let U =
{f/,},e/ he an open covering of X. Then we have:
(1) Suppose the cohomology group H"[U,.J) defined by an infective resolution
of J vanishes for each n > 0 and for each s = (/() i„i) € /"H'. where in > 0. too.
is arbitrary. Then H"[!d.J) = H" [X.J) for each « > 0.
(2) Assume that for each open covering 23 of X there exists an open covering
2B which is a refinement of 23 and which satisfies the condition above in (1). Then
H"[X.J) = H" [X.J) for each n > 0.
Proof. (1) Look at the double complex C = {C'"[ii.Jr").dl".S"} which we
gave right after Corollary 2.20. Since ,f" is a scattered sheaf. H'f" = (0) (for each
m > 0 and each n > 0). The hypothesis on U and J also implies H"'/" = (0) (for
each m > 0 and each n > 0). Thence. Hp(f) = H^K*) (for each p > 0) by
Lemma 2.21. Since Hi'[f) = H>'[X.J) and H''{K*) = H''[iX.J). we obtain an
isomorphism as stated in (1).
(2) This part is clear by the definition of H'"[X.J). □
In order to compare two cohomologies {//"(X. •)}«>() and {H"[X. -)}»>o we can
employ Lemma 2.17 in several cases. By Lemma 2.19 and Corollary 2.20. H"[X. .J) —
(0) [n > 0) for an injective .^-Module Jf. Hence, it suffices to check whether or not
an exact sequence of .^-Modules
0->7|Aj:ij,^0
gives rise to an exact sequence of Cech cohomology groups:
0 — H{)[X.J) £ Hn[X.J2) C H()[X.J)
^ Hl[X.J) ^ H\X.J2) ^ H\X.J) -^---
^' H"[X.J) ^ H"[X.J) ^ H"[X.J)
^H"^[X.J\) -+■■■ .
This holds if X is a paracompact topological space. (Recall that X is paracompact
by definition if A' is a Hausdorff topological space and each open covering U of X
has an open covering 23 which is a refinement of il and is locally finite. We say that
an open covering 23 is locally finite if. for each x G X. there is an open neighborhood
W of a which meets only finitely many open sets belonging to the covering 23.) The
following result is proved in [1] though we do not make use of it in the present book.
Theorem 2.23. If X is paracompact. we have H"[X.J) = H"[X.J) [all n > 0)
for an arbitrary s4-Module J.
In order to ultilize cohomology theories effectively, we often use the spectral
sequences which we now explain.
Definition 2.24. Let R be a commutative ring. A collection of /^-modules and
tf-homomorphisms E = {£T;.d'r'-"-Zx{E'^1). B^[Ep). E":p.q.n.r e Z. /• > 2}
is called a spectral sequence if it satisfies the following conditions (1)-(5).
(1) E',''1 and E" are /^-modules.

s:
I. PRELIMINARIES
(2) dp,/: Ep" -> Er1'-'/-'^1 js an tf-homomorphism. and
dl"r'i-r n -di'Li = 0.
(3) Let Zr+](ED = Ker(dD and 5,,,(^) = Im^r^'"1). Then (by (2))
an /^-module inclusion B, , i (E,'''') C Z, + l (£('''') obtains, and there is an isomorphism
<": Z,M{EP-i)/Br+l(£,"■") ^Ef;V
By induction on s = k - r. we define the .R-submodules Zk{EPH) and Bk(E!l-''1)
(k > r + 1) of Ef'1' as follows. When ,v = 1. they are already defined. Suppose they
are defined up to ,v — 1. By means of the isomorphism a1,''1, we can identify E''+x
with an .R-submodule of EP'L' / B,- m {E1,''1'). With the natural siirjective homomorphism
7i: EJ''1 -> E!'-c,/Brrl(Er<). we define Zk{E',"') : = n~x {Zk{E™{)) and ^(E^) : =
7^(^-(^+,)). We set Br(Ep'r) = (0) and Zr(Ep") = £?'''. Then we have the
following inclusion relations:
(0) = Br(Ep'<) C B, . i(£f'') C ■ ■ • C Bk(£,M) C ■ ■ ■ C Zk(£f'')
C-cZ,4l(£f'')cZ,.(£f-') = £f-''.
Note here that Epq S ZA. (£^)/^-(^) (A: > r).
(4) There exist fi-submodules BX{E'^) and Z^ED of E?" such that
5,(£?■") c B^(£?*) C Z.^(£™) c Z,(E1'") (k>2).
(5) £" has a descending chain of 7?-submodules (called a filtration)
E" D ■•• D £''(£") DF'ri,(F) D •■■
such that gr/E") = ££? := Z^E^/B^E^) (n = p + q). where gr/E") : =
E/'(£")/E/,H(£").
We call {£"}„ez the ahutcment of £. We simply write the spectral sequence E
as£ = {E^.E"}.
A spectral sequence E = {£("'.£"} is called bireguiar if the following two
conditions are satisfied:
(i) Z^ (£?'') = Zk(E';c>) and B^E'^1) = ffA.(EC'') for some k > 3.
(ii) For each « there exist integers /(«) and m(«) (/(«) > m(n)) such that
fi(»)(E") = (0) and £»<(")(£") = £".
In what follows, in order to simplify the situation, we only consider spectral
sequences which are bireguiar and lie in the first quadrant. Namely, we consider a
bireguiar spectral sequence E = {Epc>. E"} such that EP'1' = (0) whenever/? < 0 or
q <0.
Lemma 2.25. Suppose a speetralsequence E = {E1,''1. £"} lies in the first quadrant.
Then we have:
(1) //'/• >p. then Bk{Epq) = (0)for each k > r. Ifr>q+\. then Epc> = Zk{Ep")
for each k > r.
(2) Ifp < 0. then FP{E") = FP~X{E")
Ifp > n. then F''{E") = Fp '' (£") = ■ ■ ■.

2. THHORY OF SHFAVFS AND COHOMOLOGIhS
53
Proof. (1) The hypothesis that EC,''1 = (0) if p < 0 or q < 0 entails that
E'r'1 = (0) for each r > 2 if p < 0 or q < 0. We shall prove the assertion (1)
by induction on s = k - r. If r > p. then Br+\(EP'!) = Im^r'""-') = (0)
and 7i: E1'1' —> E,1LI jB,^\[Epr'q) is the identity homomorphism. By the induction
hypothesis. Bk{Epi!) = n-\Bk{E™x)) = (0) for each /c > r + 1. If r > q + 1. then
Zr+i(£?■*) = Ker(dD and Ep+rii~, + { = (o). whence Zr+1(£fv) = £'''". For each
k > r + 1 the induction hypothesis entails Zk{EP'L<) = n~] {Zk{Ep'^)) = ^(£^,) =
E'r'1' because n is the surjection from Ep'1' onto EP1X.
(2) Suppose p < 0. We have £/'-'(£")/£/'(£") = Z00(£^l■'')/500(£^1■',)
Co + q = « + 1). and Z00(£^~'''/) is an .K-submodule of £2'~''''. Since /7—1 < 0.
we see £?"''' = (0) and. accordingly. Z0O{Ep_~['l')/B0O{Epfl11) = (0). Therefore.
F''-l(E") = FP{E"). By a similar reasoning, we have Fp-]{E") = F''-2{E")
Suppose /? > n. Then F>'{E")/F>'+]{E") * Zx{Ep_x,)/Bx{Ep2")(p + q = «). and
£^ = (0) since q = « - p < 0. By the same reasoning. £''(£") = F'n[(E").
Similarly, we have £'HI(£") = £/>+2(£") = • • •. □
Lemma 2.26. If a spectral sequence E = {EP ''.£"} « biregular and lies in the
first quadrant, then there exists an exact sequence
0^E\0XEl^E02A''^Ef0XE2.
where ip stands for the natural homomorphism
ep.o ^ Ef/B>(Ef) ^ Ef -> Ep0 -> • ■ ■
— ££? = F''(E") ->■ E" = EP.
and 7i i stands for the natural homomorphism
El = F{)(El) ^ F°{El)/Fl{E]) * £^1 ^ Z?(£201) -^ £2al.
Proof. Since £,U) = £] ° = ■ ■ • = £^°. /, is injective. Note that FP^{EP) = (0)
by (2) of Lemma 2.25. By the definition of n\. we have also Kerrei = E\ °. Similarly,
by the definition of tt,. we have Im^i = Ker^01. Since £~° = £42() = £^. we have
Ker i2 = B} (£22 °) = Im j" '. D
Lemma 2.27. If a spectral sequence E = {£("'.£"} is biregular and lies in the
first quadrant, we have:
(1) lfEPM - (0) for each q > 0. then £^° = £'' for each p > 0.
(2) //£?" = (0) for each p > 0. /Ae/i £^ = E" for each q>0.
Proof. (1) By the hypothesis. EPH - (0) for each q > 0 and each r > 2. Hence.
Em s F''{E!Hli)/f''^(El'+l') = (0) for each q > 0. So £'' = £°(£'') = ••■ =
£''(£'') = £^°. On the other hand, since Epf S £^''0 S ... S £^«. we have
EpS = £''. We can prove the assertion (2) in the same fashion. □

54
I. PRELIMINARIES
By EP2H =4- E" we denote a spectral sequence E = {E1,'1'. E"} which is biregular
and lies in the first quadrant. This is because all the modules £'"' and ££.''. except
for the boundary operators df'1' and the filtration {FP{E")} of E". are determined
by £■?''■ In Definition 2.24. we can consider a spectral sequence of ^-Modules over
a ringed space (X.s£). where E'f'' and df''1 are J3/-Modules and J3/-homomorphisms.
Then Lemmas 2.25-2.27 hold as they are. In this book, we make use of the following
two spectral sequences.
Theorem 2.28. Let <t> = (/. <p): (X. si) —> (Y. 33) be a morphism of ringed spaces,
and let SF be an sf-Module. Then there exists a spectral sequence
EV = W'{Y.Rl'j\&) => H"{X.^)
which is biregular and lies in the first quadrant.
Theorem 2.29. Let <t> = (f.ip): (X.sf) -> (Y.33) and *¥ = {g.y/): (Y.33) -^
{Z.W) be morphisms of ringed spaces, and let & be an si'-Module. Then there exists
a spectral sequence
E>:f> = Ri'gM'f^) => Rn{g.fu^)
which is biregular and lies in the first quadrant.
The spectral sequences in Theorems 2.28 and 2.29 are called the Leray spectral
sequences. In proving the existence of these spectral sequences, we use a more general
theorem [2. Theorem 2.4.1] and the following simple fact:
If 5^ is a scattered ^"-Module, then f*9~ is a scattered
^■-Module.
In the present book, we will not go into further details of the existence proof.
Finally, we quote [2] as a general reference related to the contents of this chapter,
which is now a fundamental reference concerning abelian categories and their
applications. It is now appropriate to call this a classical reference. The readers are
advised to have a look at it.
1.2. Problems
1. In what follows. {X.si) is a ringed space.
2. For an exact sequence of J3/-Modules
verify the following assertions.
(1) If.!F is finitely generated and^ is quasicoherent. the^ is also quasicoherent.
(2) If %? is finitely presented, i.e.. for each v e X there exist an open
neighborhood U. positive integers p.q. and an exact sequence of ^-Modules
{sZi>y< -► (srfLy -> 3?,- -> 0. and if $> is quasicoherent. then "S is also
quasicoherent.
3. Find an example of a homomorphism of J3/-Modules ip : & —> "S for which the
presheaf %'. as defined in Lemma 2.6. is not a sheaf. (If we use a result to be
explained later, we can find the following easy example. Let C be a nonsingular

2. THI.ORY OF SHF.AVFS AND COHOMOLOOII.S
55
projective curve and let P. Q be points of C. Then we have an exact sequence
of <fc-Modules
0 -> &C\-P -£>)-> <9c -^ <9p * &q -+ 0
for which r{C.if({-P - Q)) = (0) and T(C.ifc) = k ( = the ground field) entail
T(C.g") - k though "f" = ^ ¢)^ and r(C."%") = k q*k. Hence g" 7^ "g".)
4. Find an example of .(/-Modules 9~. "§ for which the presheaf 2P defined by
&{U) = ^{11) igjtffu) &(U) is not a sheaf. (We find the following example
again using the results to be explained later. Let P be a point on the projective
line C = P1. let ^ = @C(-P)- and let 5? =@C{P). Then & ®5? =&C- though
r(C^) = r(C.<fr(-/>)) (¾ r(C.<fr(P)) = (0)(¾ A- = (0). Hence. ,9° f- "&>.)
5. Show that if 3~ is a scattered J3/-Module. then each term g"'(U. SH appearing in
the Cech resolution of & is a scattered sheaf as well.
6. Verify the following assertions step by step.
(1) For an ^-Module &. define an if-Module S{^) by S(§r)(U) = Fiver-^
and /;rr'- Fiver ^ ~* Fiver ^ (the natural projection). Then S{9~) is
a scattered sheaf. Moreover, a homomorphism 77: ,^ -^ <SW) defined by
.v e •?"(£/) >-> Fiver ^ e ^(-^)(^) is injective.
(2) We can define a scattered resolution of 9~
byS0{9-)=S{9r).S\9-)=S{S*(&)/ri{&)) <£"(5H=<£(<£"~ WA*"-2
{S"^1^))) We call this the standard scattered resolution. This gives
rise to a complex
0-+ r(x.<s°(9-)) £ r(x.<s\&)) £■■■ C r(A\^"(^)) £ • ■ • .
We denote by H"{X.!?) the cohomology group of this complex.
(3) Let
0 ^ jr .£, &o £ ^.' £ ... ''£' ^» £ ...
be an arbitrary scattered resolution of &. i.e.. an exact sequence such that
each term except for & is a scattered J^-Module. Let H"(X .^) denote the
cohomology group of the complex
0-+r(x.^())£r(A-.^1) ^../£' n*. .»")£...
which the above scattered resolution of Ogives rise to. Then we have H" {X.S^)
= H"{X.&).

5(1
1. PRHLIMINARIBS
(Hint: We can define naturally the following commutative diagram:
0 0 0
0-
-m{)-
0 ^S{){?)^^S"{.9ZQ)
o—^ sx{9-)—^sx{m°) —
s„ I Y
>mn
o—> S" (&■
■S"
MU H"
Y
where all rows and columns are exact sequences. Applying the functor V(X. •)
to this diagram, we obtain a double complex as we did right before Lemma
2.21. Next by utilizing the arguments in (2) of Lemma 2.19 and Lemma
2.21. show that two cohomology groups H"(X.^) and H"(X.!F)
coincide with each other.) {H"{x.')}„>0 is a cohomology functor on a ringed
space (X.s/). By making use of (1) of Lemma 2.19. show that H"(X. •) =
H"(X.-) for each n > 0.
(4) Every ^/-Module 9~ can be regarded as a Z-Module. i.e.. an abelian sheaf.
Denote by {H"b(X. -)}„>o the cohomology functor on a ringed space (X. Zx)
defined by means of injective resolutions. Show that H"(X.SF) = H^(X.^)
for an arbitrary ^/-Module &~. (Hereafter we identify these cohomology
groups.)
(5) (Hint: Note that a scattered ^-Module is a scattered Z-Module. Using this
remark, show that H" {X. &) = H"(X. &) = H'^{X. &) ^ H^{X. &). Here
{H"t,(X. •)}„>() is the cohomology functor defined by means of resolutions
by scattered Z-Modules.)
7. Suppose for an ^-Module & there exists a closed set Y such that Supp.9" : =
{.v £1:^, f (0)} c Y. Show then that H^X.P) = H'^Y.^W) for each
«>0.
Hereafter we assume that A' is a noetherian space (cf. Definition II. 1.5).
8. Let \JF,\ i G /} be an inductive system of ^/-Modules (or abelian sheaves) indexed
by a directed set I. Define a presheaf 2P by &>{U) = lim ^(U). Show that 3?
is an J^-Module (or an abelian sheaf). We denote 2P by lim ^ and call it the
limit of the inductive system \JF,\ i € /}. Show that if 5^ is a scattered sheaf for

2. THEORY OF" SHFAVr.S AND COFFOMOLOG1F S 57
each i e I. then lim .% is a scattered sheaf. Show also that //"(AMim .%) =
lim WiX.^i) (for each n > 0).
(Hint: In order to prove the first assertion, prove and use the following result:
A closed set and an open set of a noetherian space are also noetherian spaces.
Let U be an open set of a noetherian space X. and lei U = { C/„}„£^ be an open
covering of U. Then there is a finite open covering of U consisting of finitely
many members of 11. In order to prove the last assertion, use the fact that lim
is an exact functor, i.e.. the limit of exact sequences of abelian sheaves indexed
by I is an exact sequence.)
9. If the Krull dimension of X (cf. II.3.5) equals n. then we have H'iX.F) = (0)
(all i > n) for any abelian sheafs. Verify this assertion by induction on n
following the next steps one by one.
(1) If/j — 0. then Iisa finite set with discrete topology. Then H'{X'.!¥) -= (0)
for each i > 0.
(2) Let X be a decomposition of X into irreducible components. Let p,". X/ -+
X be the natural embedding for 1 < ./ < r. and let &~[x, - f)^ be the
inverse image of 9~ by tpr Since X,- is a closed set of X. we can extend
■?\x, to a sheaf,?} on X by setting &\\x-x, - (0)- Then there exists a
natural injective homomorphism y/: & —> 0^--,5^. Let 5* = Cokeri//.
Then Suppg? c \J;/k X,- n Xk(-~ Y). The Krull dimension of Y is less
than «. By the induction hypothesis. //'(A\®;. ,.?",) = ®J , H^X ,.^,)
for each i > n. Hence, it suffices to show that H'{Xr!?j) — (0) (for each
i > n and each /. 1 < / < r). We may. therefore, assume that X is irreducible.
(3) Let Y be a closed set of X. and let U — X — y, Let Z^ be a constant
sheaf with (ZA')V = Z (for each .v e -V). and let Z>- denote, by the abuse of
notation, the constant sheaf Zy on Y extended to X with Zy\i = (0). We
can define a natural surjective homomorphism ,t: Z,t< —> Z>- as follows: Let
F be an open set of X. If V n K 7= 0. then /t(F) - id/ and if fnf = 0.
then tt(K) = (0V Denote by Z,; the kernel of n. Then Z,-(V) = (0) if
F n T 7= 0. and Zt.-( K) = Z if V n 7 = 0. For an element .v G jT(t/) define
a homomorphism of abelian sheaves </\ : Zt - >.? by setting tps (K) = (0) if
V r Y 7= 0 and </>, (K)( 1) = /^t;(.v) if V n T = 0. Let,?", = </>,(Zr). Then .%
is a subsheaf of .?". and Supp(Kei'y\) is a proper closed subset of X provided
v f 0. Furthermore, there exists a surjective homomorphism of abelian
sheaves p: (J)/€/ Z[.-, - * !? for a suitable set I. Write «?} = p{ZV]). Then
-?" = U/f/ -^/ = lim 5*", ■ So H'{X. SO = lim //'"(A'. ?)). In order to show
that//'(A'. ^) = (0) for each i >n. it suffices to prove that // '(A'. Z,;) = (0)
for each i > n.
(4) Using the fact that Zx is a scattered sheaf on X and using an exact sequence
0 -^ Z[■ -> Z.v -- Zy — 0 (K :- X - £/). verify that H'(A\ Z(■) = (0) (for
each / > «).
(5) With the notations of Lemma 1.2.9. verify by following the next steps that
R''/,{/*% $>?) - % $ R?ft& [p > 0) for an j*-Module 9' and a locally
free ^-Module g\
(6) For a ^-Module rS. there exists a natural ^-homomorphism a.: 3 i£> f x9~ —
f\\f*W C;i.?) such that a is an isomorphism provided "3 -%.
(Hint: First of all. show the existence of a by making use of Lemma

58
I. PRELIMINARIES
1.2.9. In the case 5/ = "S. a\i is an isomorphism if U is an open set of Y
such that W\c is a free 3S\L-Module.)
(7) Let 0 —> 9~ —> .y° —> ^1 —> ■ ■ ■ be an injective resolution of 5^ Then
0 -> /*f o$ 9~ -+ f*% » J^° -> /*r 50 J^i -> • ■ • is an injective resolution
of/^0¾^.
(8) By making use of (1) and (2) above, show that f 50 tf''/*^ = R''ft{f*% <X
5*") for each /? > 0.

Part II
Schemes and Algebraic Varieties

CHAPTER 3
Afflne Schemes and Algebraic Varieties
This chapter is an introduction to more general concepts of schemes and algebraic
varieties.
Let R be a commutative ring. We denote by Spec(7?) the set of all prime ideals
of R and call it an affine scheme defined by R (or with coordinate ring R). We read
Spec(ft) as spectrum (or spec, for short) of R. For a subset E of R. let V(E) =
{p e Spsc(R):E c p}. and let D(E) = Spsc(R) - V(E). With the ideal 1(E) =--
J2/GERf generated by E. clearly V{E) = V{/{£)) and D{E) = D(l(E)). If £ is
a set consisting of a single element E = {/'}. then we write V({/}) = V(f) and
£>({/}) = £>(/). Obviously. V{\) = 0 and K(0) = Spec(tf)- Correspondingly, we
have /)(1) = Spec(fl) and /)(0) = 0. We have then the following result.
Lhmma 3.1. (1) For ideals I.J.I,. (/1 e A) of R. we have the relations below:
(i) V{I) U V{J) = V(ir\J) = V{IJ). where IJ is the ideal of R generated
by {xy.x G I.y G J}.
(") H;ga F(y') = K(E/Ga//)- ,v/lm' E/eA7/ « f/7t' '*"/ "/'^ generated
by I, (A € A),
(iii) F(0) = Spec(fl). K(/0 = 0.
(iv) V(I) C F(J) ?/<m</ (w/>> if VJ C v7/.
(2) /w f/ic complement £>(/) «/ K(/). u'r /ww f/u> relations below:
(i) D{I)C\D(J) = D(I r\J) = D(IJ).
(ii) U;eAD(/J=i)(E;eA^
(iii) £»(0) = 0. £»(7?) = Spec(fl).
(iv) /)(/) =u/e/ W).
(v) /w p G /)(/). f/zcw is an element f G R such that p e /)(/) C /)(/).
Proof. (1) (i) Note that F(/) C V{I) if/ C J. Since /JC/niC/ (and
also C J). V{I) U f (/) C F(/ n J) C K(/J). If p G K(//). then / C p or J C p.
Hence. V{U) C K(/) U F(J).
(ii) Since /, C £/eA/,. ^(E;£a^) ^ fl;.GA ^)- If P G PUa ^(';)-P'2 /,
for each ieA. Hence, p D J2;€A I,-
(iii) Obvious, (iv) follows from (1.1.24).
(2) (i). (ii). and (iii) are equivalent to (i). (ii) and (iii) of (1). respectively.
(iv) If/ e I. then /)(/) C /)(/). Hence. U/e/^C/') C /)(/). If p 75 /. then
/ £ p for some / G /. Hence, p G £>(/). Namely. /)(/) C |J/£/ £>(/).
(v) has been proved in (iv) above. □
Definition 3.2. Lemmas 3.1 shows that Spec(Z?) admits a topology with V{l)
(£>(/). resp.) as a closed set (an open set. resp.) when / ranges over all ideals of
R. With respect to this topology. {/)(/):/ e R} forms an open base. We call this
61

62 II. SCHEMES AND ALGEBRAIC VARIETIES
topology the Zariski topology, and we consider hereafter that Spec(ft) is given the
Zariski topology.
The Zariski topology on Spec(7?) is a T^-topology. though it is not a T\ -topology.
In fact, if pi. p2 £ Spec(-R) satisfy pi ^ pi then any open neighborhood D(I) of p2
necessarily contains p|.
Definition 3.3. Let tp: R —> S be a homomorphism of commutative rings, let
X = Spec(^). and let Y = Spec(S). Define a mapping of sets f: Y —> X by
assigning /(q) := ip ~'(q) to q £ Y. We denote / by "ip and call it the morphism of
affine schemes associated with ip.
Since f~\V{I)) = V{tp{l)). f is a continuous mapping with respect to the
Zariski topology, where ip{I) = {ip(x);x e /}. Furthermore, if ip: R\ —> R2 and
i//: Rt —> R^ are homomorphisms of commutative rings, then we have "{i// ■ ip) =
Lhmma 3.4. (1) Let I be an ideal of R. and let n: R —> R/I be the natural
residue homomorphism. Then the mapping "n: Spsc(R/I) —> Spec(7?) gives rise to a
homeomorphism between Spec(R/I) and the elosedset V(l) with the induced topology.
(2) Let S be a multiplicatively closed subset of R with 1 £ S and 0 ¢. S. Then
Spec(5'] R) — Pl/e.v -^(./)- «rtt/ r/ic topology on D/e.v ^(/) induced by the Zariski
topology qf'Spec{R) coincides with the Zariski topology ofSpec(S~lR).
Proof. The correspondences of ideals given in the proof of (1.1.13) yield the
correspondences of the same kind between prime ideals. It is straightforward to
prove the assertions if this remark is taken into account. □
In order to describe topological properties of affine schemes, we have to prepare
with some definitions concerning topological spaces.
Definition 3.5. Let X be a topological space.
(1) A closed subset F of X is reducible if F — F\ U F2 for proper closed subsets
F\ ■ F2 of F. We call a closed subset F irreducible if it is not reducible. In particular,
if F is irreducible, then F = F\ U F2 with closed subsets F\ ■ F2 of F implies F = F\
or F = Fi.
(2) X is called a noetherian space if given an arbitrary descending chain of closed
subsets
X D XQ D X\ 3 • ■ • D X„ 2 • ■ ■
there exists N > 0 such that X„ = XN (for each « > N).
(3) Let
Z.CZ.C.C/
be an ascending chain of irreducible closed subsets of X. We define the length of
the sequence to be r. We define the Krull dimension of X to be the maximum of the
length of ascending chains of irreducible subsets of the above kind, and denote it by
dimX. We could have dimZ = oo.
(4) X is quasicompact if given an arbitrary open covering It = {£/,},g/ of X.
there exist finitely many members £/,-, U,n of it such that X = U,t U Uh U ■ ■ • U U,n.
Lemma 3.6. (1) A closed set and an open set of a noetherian space are noetherian
spaces with respect to the induced topologies.
(2) A noetherian space is quasicompact.

3. AFF1NF SC'HI-.Ml'S AND ALCihBRAIC VARI1 Til S
6¾
(3) X = Spec(ft) is quasicompact though it is not necessarily a noetherian space.
Proof. (1) and (2) are apparent. We shall prove (3). Let {D{I,):a e A} be
an open covering of X. Then Y1>ga^ = ^ by Lemma 3.1. Hence. 1 = YH-\Ll'
with a, £ /;., and />,• £ A. This implies that D(a:) C £)(/,.,) and X = \J" lD{a,).
Consequently U'/_, £>(/,„) = X. ' □
Lemma 3.7. Let X = Spec(7?). Then we have
(1) For a closed subset V(I) of X. V{I) is irreducible if and only if \/l is a prime
ideal.
(2) For p £ X. V(p) — {p}. where {p} denotes the closure of the point p. We then
say that p is the generic point of the irreducible closed subset V{p).
(3) p £ X is a closed point if and only if p is a maximal ideal of R.
Proof. (1) We may assume / = \/l. For the "only if" part, suppose ah e I. Set
J = 1 +aRnndK = I +bR. Then JKQI. Hence. V{J)U V{K) = V{JK')D V(I).
V{J) C V{1). and V(K) C K(/). So F(/) = K(J) U V{K). Since F(/) is
irreducible, either V(I) = V{J) or V(I) = V(K). So I = sJJ or / = ^Z. Hence.
a £ I or h £ I. Namely. / is a prime ideal. For the "If" part, suppose there exist
ideals J. K such that /5 I.K D/. and F(/) = l'(y)UF(4 Then V{I) = V(JK).
whence JK C /. Since / is a prime ideal. J C I or K C I. So J = I or K — I.
Namely. V{J) = V(I) or V{K) = V{I)._V[I) is therefore irreducible.
(2) By the definition of the closure {p}. we know {p} C F(p). Moreover, if
p £ V{I). then K(p) C V{1). Hence. V{p) = {p}.
(3) {p} — p 7^ 0 if and only if there exists some q £ Spec(fl) with q ^ p. Hence,
the assertion follows. □
Corollary 3.8. For X = Spec(fl). dimA' = K-dim« {see (1.1.16) for the
definition of K- dim /?).
Proof. Since the set of irreducible closed subsets of X is in one-to-one
correspondence with the set of prime ideals of R. an ascending chain of irreducible closed
subsets of X corresponds bijectively to a descending chain of prime ideals of R. The
result follows from this fact. □
Recall the one-to-one correspondence between the set of closed subsets of X =
Spec(7?) and the set of radical ideals of R (Lemma 1.7). This correspondence is
inclusion-reversing. In particular, if R is a noetherian ring, then X is a noetherian
space, though the converse does not necessarily hold (Problem II.3.1).
Lemma 3.9. A closed subset F of a noetherian space X is uniquely expressed as
a finite union of irreducible closed subsets
F = F\ U • ■ • U F,.. F, : an irreducible closed subset of F.
F g Ft U • • • U Fj U • • ■ U F, { for each i. 1 < / < r).
Each of F\ Fr appearing in the finite union is called an irreducible component
of F. When R is a noetherian ring. X = Spec(ft) and F — V{I) with a radical ideal
I. the decomposition of F into irreducible components corresponds to the prime divisor
decomposition (/.1.23).
Proof. Let S be the set of closed subsets F of X which are not expressed as the
finite unions of irreducible closed subsets. We shall show that the assumption & ^ 0

64
11. SCHEMI S AND ALGEBRAIC" VARIETIES
leads to a contradiction. Introduce in 6 a partial order F > F' by F C F'. Since
A' is a noetherian space. & is an inductively ordered set. Then by Zorrfs lemma,
there exists a maximal element, say F(). in &. Since Fq is an element of &. Fo is
not irreducible. Hence. F0 is reducible, and F0 is expressed as a union of closed
subsets Fo = F\ U F2 with F, C f() (/ = ].2). Since F0 is a maximal element of 6.
F, ^ © (i = 1. 2). So F, is expressed as a finite union of irreducible closed subsets:
F, = F,, U F,2 U ■ ■ ■ U Fir, (i = 1-2). Then F0 = \Jj , U'/-i ^/- which is a finite
union of irreducible closed subsets. This contradicts the choice of Fo. It follows that
e = 0.
Now let F be an arbitrary closed subset and express it as a finite union of
irreducible closed subsets F = F\ U • • • U F,-. where no inclusion relations exist among
{F\ F,.}: this is always possible. Suppose F = G\ U • ■ • U G, is another expression
of F as a finite union of irreducible closed subsets among which there are no inclusion
relations. Then we have
G, = (G, nF,)u---u(G, nF,).
Since G\ is irreducible. G\ = G\ n F, for some /. We may assume / = 1. Then
G| c F]. Apply the same argument to a decomposition
F, = (F, nG,)u---u(F, nGA)
to obtain Fi C G^ (for some /). Then G\ C G;. whence / = 1. So Fi = G\. The
above argument implies that r = ,v and G, = F„(/) (for each /. 1 < / < r) for some
permutation a of {1.2 r). Namely, we conclude that the way of expressing F
as a finite union of irreducible closed subsets is unique.
When X = Spec(7t"). F = V(I). and I = \fl. then each irreducible component F,
is expressed as V(,Pi) with p; e Spec(7t"). and F being expressed as F = F\ U ■ ■ ■ U F,
is equivalent to I being expressed as / = pi n ■ ■ • n p,-. The last decomposition is
nothing but the prime divisor decomposition of /. □
Let X be a noetherian space, and let Z be an irreducible closed subset of X.
Then Z is also a noetherian space. Considering an ascending chain of irreducible
closed subsets of X which passes through Z so that Z is one of the irreducible closed
subsets in the ascending chain, we know that dimZ < dim A'. If Z is reducible and
Z = Z| U ■ • ■ U Z, is the decomposition into irreducible components (the irreducible
decomposition, for short), we have by definition that dimZ = sup1<(<, dimZ, and
dimZ,- < dimZ for each /. So dimZ < dimZ again. Suppose anew that Z is
irreducible. Consider all possible ascending chains of irreducible closed subsets of
X which start with Z.
Z = F0C/r, C...Cf,
The maximum of the length s is called the codimension of Z in X and is denoted
by codim^-Z. If Z is reducible, we define codim^ Z = inf|<,<,. codim^ X,. where
Z = Z] U • ■ • U Z, is the irreducible decomposition.
When R is a noetherian ring. X = Spec(ft). and Z = V{I). then coding Z is
equal to the height ht(/) of/ (cf. (1.1.30)).
Our next purpose is to define a local ringed space structure on Spec(Tt'). We
need one more lemma.
Lemma 3.10. Let X be a topological space, and let {U0:a £ A} be an open base
of X. Suppose a set (or an abelian group or a commutative ring, resp.) 3^(11 a) is

3. AFFINF SCHFMFS AND ALGFURA1C VARIhl'IFS
65
assigned to each Ua (a e A) and the restriction morphism ppn; &>(U„) —> 2P{Up).
which is a mapping of sets (a homomorphism of ahelian groups or commutative rings,
resp.) is defined to every inclusion Up C U„ in such a way that />„,, = id#(t j and
/;;« = I'Ji " />/<<> whenever Uv C Up C Un. Define £?y = lim £P(Un) for each
x e X. Then there exists uniquely a sheaf of sets (or ahelian groups or commutative
rings, resp.) 5F on X such that &~x = 2P\ for each x £ X.
Proof. Set 3P = Ti^x^0^- For each a e A and eacn v e ^>(u«)- define a
subset V(a.s) of & by {sx:x £ Un}. where ,vv is the image of .v in SPy. Suppose
V(a.s)n V(li.t) ^0. Then for ,s\(= ?J £ K(a..v)n F(/?.f)- there exist £/.. (y e ^)
and w e @{U..) such that ,v € U-. c £/fV n 6), and ,sA e f (/. m) C V(a. s) n L (/?. ?)■
Hence, we can define a topology on 52 for which {V(a.s):a € ^-.v € ^(C/,,)}
is an open base. Define n: 3P —> A' by 7r(.vY) = -v. Then & and A' are locally
homeomorphic to each other under n. Now define .9~(U) = {a: U —> ^: u is a
continuous mapping with 7i • ct = id^,■} for any open set U of X and the restriction
morphism/;rt;: fF(U) —>&~(V) in a natural fashion. Then^ is a sheaf (cf (1.2.5)).
We shall show that 3~ is uniquely determined. Let 'S be a sheaf as required in the
statement. Then the sheafifying spaces 5F and 'S of SF and 'S are homeomorphic to
2P. Hence. & is homeomorphic to "S. So fF is isomorphic to ^ as sheaves. □
When we regard X = Spec(7?) as a topological space, we denote its points by
x.y.... and the corresponding prime ideals of R by pv. p, For .v £ X. we
denote by Rx the quotient ring RVx of R with respect to a multiplicatively closed
subset R - pv.
Recall that {£)(/): / e /?} is an open base of Spec (ft). To each D(f) we assign
a commutative ring Rf = /?[/"']• We then have the following result.
Lemma 3.11. (1) IfD(g) C£>(/) there is a natural ring homomorphism px / : Rf —>
/^, satisfying the conditions
(i) /;// = id/?,.
(ii) /;/,/ = /¾ • pKf provided D(h) C D{g) C £)(/).
(2) Foe «/f/i a- € A\ let Ux = {£>(/):/ £ pv}. Then we have
lim Rf = /?.,.
Proof. (1) We have the following equivalent conditions: D(g) C £>(/) <=>
^(#) 2 ^(/) <=> sfgR^ \fTR<$g" — fu for some « > 0 and some a £ R. So define
pKj by A,y (c/.f) = [car)/g>". Suppose Z>(A) C/)(g)C £)(/). Then #" - /a and
h'" = gh for some n.m > 0 and some a.h £ R. According to the definition, we have
ph,(c/fr)=cu'b'"/h>"<" ■MdphH-pxf{clfi)=phf,([ca<)lgi'-) = {{ea')h'")lhF'"n.
Sop,;/ = phg -ptf.
(2) If £)(/) e C/x. define a ring homomorphism <//y : 7?/ —> /?v naturally by
¥f(c/f) = <'//'■ If * 6 ^0?) = -0(./)- we have (///■ = t//,, •/>*'/■ Hence, there
exists a ring homomorphism t//: lim Rf —> 7?Y such that i/y = ^ 'If- where
rif \ Rf —> lim /?/ is the natural ring homomorphism associated with the
inductive limit. Here ^ is surjective because any element of Rx is expressed as a/f
with a. f £ R and /' ^ p,. Take an element of lim Rf which is represented
by an element c/f of Rf. Suppose y/{c/ /'') = r//' = 0. Then rg = 0 for some

66
II. SCI II MIS AND ALG1.BRA1C VARII TII.S
g ¢p^■ Therefore. ¥>vf.f (r/f) = (''#')/(#/)' = 0- So y/ is injective. Hence, y/ is
an isomorphism. □
By virtue of Lemmas 3.10 and 3.11 there exists a sheaf of commutative rings R
on X such that Rx = Rx for each ,v e X. Since Rx is a local ring. (X. R) is a local
ringed space. We call R the structure sheaf and denote it by @x-
Let M be an ft-module. By assigning an ft/-module Mf := M &/? Rf to Z)(/).
we can define by Lemma 3.10 an ^-Module M such that Mx = M -g,R ftv for each
x £ X. Given an ft-homomorphism v. M —> N. we assign an ft/-homomorphism
r/ : A// —> A'/ to D{f) defined by Vf{.\//'') = r{.\)/f. Hence, we obtain a
homomorphism of (¾-Modules iJ: M —> /V. To an exact sequence of ft-modules
(1) O^LiM^iV^O
there corresponds an exact sequence of $*-Modules
(2) 0-> 1-^ M -A A' -^0.
In fact, the sequence of stalks at .v £ X
(3) 0 -► Zv ^ Mx -^ JVV -> 0
is nothing but an exact sequence of Rx -modules
(4) 0->Lv^MvAwv->0.
where we write Lx = L <£># ftv. ux = w K)R ftv. etc. Since the functor (•) ®ft ftA is
an exact functor, i.e.. a functor sending an exact sequence to an exact sequence, we
know the sequence (4) is exact. Hence, the sequence (3) is an exact sequence. This
implies that the sequence (2) is an exact sequence.
An arbitrary ft-module M is a holomorphic image of a free ft-module. Namely,
there exists a surjective homomorphism tp: R1 —> M. For example, we take as /
the underlying set of M and. with a free basis {e,„:m e M} of RM. define ip by
ip(cm) = m. Hence, we have an exact sequence
RJ ^ Ri 1, M ^ 0.
By the above remark, we obtain an exact sequence of ^-Modules
ffxtff'x^M-, 0.
Therefore A/ is a quasicoherent ^--Module. We shall prove the following basic results
on ^-Modules over an affine scheme X.
Theorem 3.12. Let (X .@x) be an affine scheme defined by a commutative ring R.
Then we have:
(1) For an R-moduIe M and f e R. M{D(f)) = Mf. In particular. M{X) =
M(D(l)) = M an<10x{X) = R.
(2) For R-modules M and N. the correspondence u h-> u gives rise to an
isomorphism HomR{M.N) = Honvv(A/./V).
(3) For an R-module M. M is a quasicoherent 3'x-Module. Conversely, an arbitrary
quasicoherent (fy-Module is isomorphic to an (fx-Module M for an R-module M.
Namely, the category of R-modules and the category of quasicoherent @x-Modules
are equivalent to each other.

3. AFF1NE SCHEMES AND ALCiEBRAIC VARIETIES
67
(4) Suppose R is a noetherian ring. Then for a quasieoherent ffy-Modulc EF. &
is coherent if and only if the R-module M = S^iX) is finitely generated. In particular.
&x w u coherent cfx-Module.
(5) For R-modules M and N. M &,*, N = (M &« N)~ . Furthermore, if M is
finitely presented. {M.N) = ftomR{M.N)~~.
Proof. (1) Suppose D(g) C £>(/). Then since g" = fa for some n > 0 and
some a e R. we have Mx = {M f-)K. So M\Dif) = (M/). Moreover. £>(/) = Spec R/.
Hence, in order to show M(D(f)) = Mf. we may assume / = 1 by replacing R
and M by Rf and Mf. respectively. By the construction of M in Lemma 3.10. there
exists a natural isomorphism cp: M —> M{X). We have only to show that ip is an
isomorphism.
(i) Injeetivity of p. Suppose cp{z) = 0 for z e M. i.e.. zx = 0 for each .v e A'. So for
each a- e A\ we have f\z = 0 for some /\ ^ pv. Let a — AnnK(z) := {a e ^: ;/r = 0}.
Then a is an ideal of R and a = ft if and only if - = 0. Suppose z ^ 0. Then p,■ D a
for some p, e Spec(7?). Meanwhile. /', e a and /', ^ p,. This is absurd. So z = 0.
(ii) Subjectivity of (p. Let £ e M(Z). For each .v e I. we can express £ = r//\
with z £ M and /\ ^ pv because Mx = MY. Apparently. \JxeX D(fs) = A\ Since X
is quasicompact. there is a finite open covering \Ji€t £)(/;) = X (I is a finite set) such
that £\o(f,) = -/ £ Mft for each / e /. For each <'. /' e /. we have £)(/,) n £>(//) =
D{fifj) and a natural homomorphism </?,,: M/i/; —> A/(£)(/,-/,-)). With r, — -,-
as an element of Mfifr <Pjj{zj - r,) = £|«(/,/,) -^1/5(/,/,) = 0- By the injeetivity of
tpij as we have shown above, z, = Zj as elements of Mfifr Here write z, = m,/'/"
with w,- e M. where we can take n independently of < £ /; this is possible because /
is a finite set. For each i. j e /. we have (/,-/,)'" (/'/w,- - /,"m,) = 0. where we may
take /-,-,- = /• (a constant independent of/./ e /). Replacing then w, by /•'»»,- and /"
by /f"1"''. we may assume /• = 0. So we have /"w,- = /"w/ f°r eacn '•■/' e '• On ^e
other hand, since |J,e/ £>(/,-) = X. we have ]T,6/ //^ = ^ Hence. £/e/ /;'/? = 7?.
So ^,-£/ /,-'#,■ = 1 for £,- £/?(/'£ /). Let w = ^ -£/ #,-w,-. Then for each i e /. we
have
/€/ \/£/ J
Therefore. £\r>u,) = v(w)lo(/,) f°r eacn ' £ ^- Since M is a sheaf, we have q = <p(m).
Therefore. <p is surjective.
(2) Define two mappings
«1 _
Hom«(M.iV) ^ Hom^-(M.A')
by O(h) = « and *P(a) = a(A'). By (1) above. a(X) is an 7?-homomorphism
from M to N. It is apparent that *P • O is the identity morphism. In order to show
that O • *f (a) = a. we have only to show it on all stalks. For the stalks, we have
O • T(a).v = T(a) ®R R, = a{X) ®R R, = av because Mx = M{X) »R Rx.
(3) With (1) and (2) above taken into account, we have only to prove that ?F =
!F{X) for any quasieoherent ^--Module 3~. Since & is quasieoherent. for each

68
II. SCHLMLS AND ALGEBRAIC" VARILIUS
.V g X there exist an open neighborhood £)(/) of a- and an exact sequence
(^1/,(/,)^(^1/,(/,)^^,/,-0.
Noting that &'x |/)(/, = 7/,. we let M(/) be the cokernel of y/(£>(/)): (7?, )J -» (7?,)'.
Then .9"|/>(/) - A/(/). Hence. <?(D(f)) = A/(/). Since X is quasicompact. there
exist a finite open covering |J ; £)(/,) = ^ and an /^/-module A/,- (/ e 7) such
that &|/>(/-, = M,. Putting M = 5s"(X). we shall show that M, = M/, for each
i e 7. Since there is a natural homomorphism a: M —+ SF. we have a natural
homomorphism of /^-modules w,- := <*(£)(/,-)): M/t —> A/,-. We show that w,- is an
isomorphism.
Injcctivity of Uj. Suppose w,-(.v) = 0 for ,v G M. i.e.. ,vv = 0 for each .v G £)(//).
Then .s\ = 0 for each 7 e / and each a e £)(/,) n £)(/,) = £>(/,-/,). Since
-^/)(/,/,) = (-^l/>(/;))f/x/,/,) - ^//1/5(/,/,, = ((A£/)/,/,)~- (////)""* = 0 as an
element of A/ = ^(£)(/,-)) for some «,, > 0. Since / is a finite set. we may assume
njj = n ( = a constant). Moreover, since (/,-)-1 G @x(D{f jj). f"s = 0 as an
element of SF{D{f,)). This holds for each / g /. Hence, f'js = 0 as an element of
M = &~{X). So .v/1 = 0 as an element of Mfi.
Subjectivity 0/11,-. Let ;,- G M,. Since Zj\D(fif- } G ^(£)(/,//)) = (A/)/,/-,.
(////)""-/ e A/, for some «,, > 0. Since (/,)-' e cfx{D{f,)). /,""r, e A/. Take
an integer « so that « > supy(«,,). Then /J'r, e A/,- for each / 6 /. Let - = /":,-.
Then r e M. So :,- = -/./,"• This shows that w, is surjective.
By virtue of the isomorphism u,. we know that M\D^f^ — -^1/),/,, for each i G /.
Hence, a is an isomorphism of sheaves.
(4) Let & be a finitely generated, quasi-coherent ^-Module and let M = ^(X).
Then there exists a finite open covering X = |J,€/ £>(./) sucri that A//, is a finitely
generated /^-module. Let {r,;.:A € A,} be a system of generators of A/,. Since
f"" --,-,. G M for some «,;, > 0. we may assume :,,. G A/ for each / g 7. A € A,. We shall
then show that {r,-;.: /6/.leA,} is a system of generators of A/. Take an arbitrary
element m G A/. Then w|/)(/,) = 5Z;6a,Wo.lf\")-\;.- Hence, with r > sup;(r,-;.).
we may assume that f.m = J2;e\ "/;-'' as elements of A/, where a'; 6 R. Since
X = U,e/ £>(/,r). we have £,€//; g, = 1 with g.efi. Som = £,'6/./¾^ =
J2,ei S/gA a'uXi:i'.- It follows that A/ is a finitely generated /^-module.
Suppose R is a noetherian ring and M is a finitely generated R-module. Given
an open set U and an efL -homomorphism ip: (&i)" —» ^"|t7. we shall show that
Ker</? is finitely generated. For this, it is sufficient to show that (K.erip)\D^-^ is finitely
generated for each v g U and an open neighborhood £)(./) of v with £)(/) C £7.
Let /V be the kernel of <p(D(f)): (R,)" -+ M,-. Then (Ker^)|/J(/) = iV. Since
7?, = /?[/"'] is a noetherian ring (1.1.14). N is a finitely generated 7?,-module.
Hence, by the first half of the proof. (Ker <p)\Dif) is a finitely generated ^(^-Module.
It should be clear that SF itself is a finitely generated ^--Module. So 9~ is coherent.
(5) M (55/-1 N is the sheafification of a presheaf U — M(U) C^,,;-, N(U). If
U = £)(/). this correspondence is given by £>(/) — A/ ®R( N/ = (A/ 55« A0/-
Hence. M (^v, N = {M (&R N)~ by Lemma 1.10. Next we have
ST^x(M.N){D(f)) = Hor^nii\M\IHf).N\DU))^HomRl{M1.NJ)
= HomR{M. N (S)r R/).

.1. AlTlNh SC'Hh'MLS AND ALGEBRAIC VARIETIES
69
By the hypothesis on M. there is an exact sequence
R" -^ R' -^ M -> 0
which entails, by an easy diagram chase.
Horn R{M.N ®R Rf)
= Ker(HomR{R'.N <g)R Rf) "U HomR{R\N®R Rf))
= Ker{HomR(R'.N) ®R R, '-U HomK{R\N) ®R Rf)
= UomR{M.N) <$>R Rf.
Therefore, we have *%V^V(M. #) = HomR{M. N)~~ by Lemma 3.10. D
Following [4]. we shall prove the next result.
Lemma 3.13. Let R be a noetherian ring and let X = Spec(/?). If I is an infective
R-module. then the associated &'x-Module I is a scattered sheaf.
Proof. Let <? = I. We shall show that the restriction morphism J*(X) -^ ^{U)
is surjective for any open set U. We note that since A' is a noetherian space U is
also a noetherian space and U is. therefore, quasicompact (cf. Lemma 3.6). First
of all. taking D(f){^= 0) contained in U. we show that the restriction morphism
J^(X) —> J*"(£)(/)) is surjective. i.e.. the natural homomorphism I —»If is surjective.
Let a = {a 6 R; f"a = 0 for some n > 0}. Then a is an ideal of R. and a = R
if and only if /'" = 0 (for some n > 0) which is equivalent to £)(/) = 0. Since
D{f) 7^ 0 by the assumption, a ^ R. Since R is a noetherian ring, a is a finitely
generated ideal. Hence. fNa = (0) for some N > 0. Now let x/f {x 6/) be an
arbitrary element of If and let x' = fNx. Define an /?-homomorphism ip: R —» /
by ip{l) = a-'. We also let a: R -> R be the multiplication by f+N: a{a) = /'VNa.
Then we have the implications: a(a) = 0 => fl+N a = 0=^a€a^fNa = 0 =>•
(p(a) = ax' = fNax ~ 0. Hence, we obtain the following commutative diagram
0 ^a{R)-^~R
/
, /
f / i//
I
where </?' and a' are the homomorphisms induced naturally by ip and a. respectively.
Since / is injective. there exists an /?-homomorphism y/: R —> / such that ip' = i// • a'.
Hence, putting y/{\) = _y. we have x' = f'^Ny. Namely, a-//' = x'/f'+N =y/\.
Here let Supp(^) = {.v 6 X;J*X ^ (0)} and let Y0 = Supp^). [If Jf is
not finitely generated. Supp(J^) is not necessarily a closed set (cf. problem II.3.2).]
Now let £, 6 <f{U). We shall show that £ = r0|t- for some :0 6 J{X). We may
assume c, ^ 0. Hence. F0 n £/ ^ 0. Take an open set £)(/) so that £)(/) C U and
D{f)r] F0 ^ 0. By virtue of the first remark. (£--i|i/)o(/) = ° for some -i e-^W-
Since U is quasicompact. /"(£ — ~iIt,') = 0 (for some n > 0) by the reasoning we
employed in the proof of (3) of Theorem 3.12 (the injectivity of «,-). We define an
/?-submodule I\ of/ by I\ = {w 6 I: f"w = 0 for some « > 0}. Then I\ is an injective
^-module by Lemma 3.13.1 below. Let J^ = /i and Y\ = Supp(J^i). Then Y\ ^ F0.
In fact. F, C F0 obviously, and F, n£>(/) = 0. and F0n£>(/) ^ 0. Meanwhile.

70
11. SCHF.MRS AND ALGEBRAIC VARIETIES
C - r||r e J*](U). If £ - r,|r ^ 0. take an open set £>(/,) so that D{f\) c £/
and D(/,) n Y\ £ 0. Then (c - (r, + r2)|r)|/)(/,) = 0 for some :2 € I\ = -f\(X).
Let h = {w € I\:f"w = 0 for some n > 0}. J"2 = h- and r2 = SuppX^). Then
^- S ^i S ^()- Since X is a noetherian space, this process terminates at finitely many
steps. Therefore. £, = z\v for some re/. □
Lemma 3.13.1. Lc/ /? be a noetherian ring, and let a be its ideal. Let I be an
injective R-module. and let J = {w e I:a"w = 0 for some n > 0}. Then J is an
injective R-module. too.
Proof. (1) First of all. note the following result:
Let / be an /?-module. Then I is an injective /?-module if and only if the natural
^-homomorphism Horn/? (/?./) —> Hom«(b./) is surjective for every ideal fa of R.
The only //part is clear. So we shall prove the (/"part. Let N be an R-module,
and let M be a submodule. It suffices to show that any /?-homomorphism <p: M —> I
extends to an fl-homomorphism y/\ N —> I. Let 6 = {(M'.ip'): M C M' Q N.
(fi'\M = ip) and define in © a partial order {M".<p") > (M'.ip') if M" 3 M'
and <p"\m' = V3'- Then © is an inductively ordered set. Let (L.p) be a maximal
element of ©. If L f- N. take r g yV - L and put fa = {a e 7?:«r e L}. Define
an ^-homomorphism a: fa —> / by a(c/) = /;(«r) for each a g fa. Then by the
assumption, there exists an /?-homomorphism /?:/?—>/ such that a — p\b. Now
define//: L + R: —> / by //(£ + a:) = /;(£) + /?(«). Then // is an ^-homomorphism
which extends p. This is a contradiction to the maximality of {L.p).
(2) Let fa be an ideal of R. and let ip: fa —» / be an ^-homomorphism. For
each b 6 fa. a"</?(/>) = 0 for some n > 0. Since fa is a finitely generated ideal.
aN(p(b) = <p(aNb) = (0) for some N > 0. By the lemma of Artin-Rees (1.1.31).
there exists r > 0 such that a" n fa = a"~''(a'' n fa) C a"-' fa for each « > r. Hence.
ip(a" n fa) = (0) for each n > N + r. Therefore, we have the following commutative
diagram.
R ^R/a"
j r^
fa >b/bna"^^7^i/
I }
v
Since I is injective. there exists an ^-homomorphism y7: R/a" —> I which extends
Tp: fa/b Ha" -^> J "-+ I. Let ^ = Tj7 • n. where n: R —> /?/a" is the natural residue
homomorphism. Then ((/ decomposes as i//: 7? —-> J ^-» /. D
The theory developed in Chapter 2 of Part I together with Lemma 3.13 yields
the following result.
Theorem 3.14. Let R be a noetherian ring, and let X = Spec(/?). Then, for any
quasieoherent &x-Moduk &. we have H°{X.9~) = Y(X.9~) and H'iX.^) = (0)
for each i > 0.
Proof. Let M = Y{X.&). and let
0 — M -> /,) -^ /| -^ • • • -> /,,-^

3. AFFINF SC'FIFMFS AND ALGFBRAK VAR1FTIFS
71
be an injective resolution of M. Then
0 —> 5* —> A) —> /, -» • • • -» /„—>•••
is a scattered resolution of 5^ Since H'{X. SF) coincides with the cohomology group
defined by a scattered resolution of JF (problem 1.2.5). we obtain the stated result. □
This theorem holds true without the assumption that R is a noetherian ring. See
[3. III. 1.3.1] for a proof.
Consider an exact sequence of P-modules
(*) O^LA M A/V->0.
The sequence obtained by taking tensor products with an P-module P is right exact,
i.e..
L %R P "-lr M »RP '''-l1' N ®« P -> 0
is an exact sequence (problem II.3.3). If the homomorphism a (g) \p is injective for
an arbitrary exact sequence like (*). then we call P a flat P-module.
Lkmma 3.15. Let R be a noetherian ring, and let P be a finitely generated R-modnle.
Then the following four conditions are equivalent.
(1) For each p g Spec(P). there exists / g P — p such that Pf is a free Rf -module.
(2) For each p g Spec(P). Pp is a free Rp-module.
(3) P is a projective' R-module.
(4) P is a flat R-module.
Proof. (1) implies (2) is clear.
(2) implies (3). Since P is finitely presented, we have
HomR(P.M)p =* Hom«p (PP.MP)
for any P-module M (cf. Proof of Theorem 3.12 (5)). Hence, if n: M —+ N is a
surjective R-homomorphism. then (nt)p: Hom«, (Pp. Mp) —> Hom/^(Pp. Np) is
surjective. So 7i*: Hornk(P. M) —> HornR(P. N) is surjective. Namely. P is a projective
Pv-module.
(3) implies (4). There is a surjective homomorphism p: F —> P from a finitely
generated free P-module F to P. Then F — P © Q. where Q = Kerp. Since F
is clearly a flat P-module. for an injective homomorphism a: L —> M. a 50 If =
{a $5 ip) (I) (a (gi Iq) is injective. whence a®lp: L®R P ^ M &R P h injective.
(4) implies (2). For each p g Spec(P). since Pp = P &>r Pp. Pp is a flat Pp-
module provided P is a flat P-module. So we have only to show that if R is a local
ring with maximal ideal m. P is a free P-module. Since P/mP is a vector space over
R/m. choose elements .V| v„ of P so that the residue classes 7q = .v, (mod mP)
form a basis of P/mP over R/m. Let Q = J^" , Pv,. Then P = Q + mP. Hence.
P = Q by Nakayama's lemma (1.1.28). We shall show that {.vi v,,} is an P-
free basis of P. We have to show that a\X] + ■ ■ ■ + a„x„ = 0 (a-, g R) implies
cij = 0 for each i. First of all. we know that m' &R P = m' P for each r > 0:
this can be seen by taking tensor products of P with terms of an exact sequence
0 —» m'' —» R —» R/m' —> 0. Let a] = a, (modm). Then ~ti\X\ + ■ ■ ■ + a„x„ = 0. Since
[x\ a7,,} is a basis over R/m. we have a, = 0 (for each i). Hence, a, £ m, for each
i. Here we show that if a, g m' for each i then a, g m' '' for each i. In fact, we have
m'/m'" ' »R/m P/mP = m'P/m'M 'P. Let <7, = a, (modm' + l). Then £"_, a>x, = 0

72 11. SC'HhMhS AND ALCihBRAlC VARIETIES
implies Yl"i..\ "< ^ ■*/ = ^. Since m' /m' H &K/m P/mP is a tensor product of two
vector spaces over R/m and {ai a,,} is a basis of P/mP. we know that each
«, = 0. So u; em''" for each /'. This implies that a, G flf>0m' = (0) for each i (cf.
(1.1.28) and (1.1.31)).
(2) implies (1). Choose elements x\ x„ of P so that {a\/1 a„/1} is an
Pp-free basis of Pv. Consider an P-free module F = X^'-i ^c'i °^ rank n ancl a
homomorphism ip: F —» P (ip(ei) = a,). Let K = Ker</? and 2 = Cokery. Then
both K and g are finitely generated. Furthermore, since Rp is A'-flat. Kp = Qp = (0).
Therefore. AT/ = Q/ = (0) for an element / of A? — p. Then P/ = Ft. So P/ is an
P^-free module. D
Corollary 3.16. Lef R be a noetherian ring, and let X = Spec(P). Then the
following conditions on a coherent @x-Module W are equivalent:
([).?- is locally free {cf. (1.2.10)).
(2) S\. /,v S^x,-flat for each x G X.
When R is an integral domain, we can characterize VyU.S'x) for an open set U
of X as a subring of the quotient field K = Q{R) of R in the following fashion.
Lemma 3.17. Let R he an integral domain, let K = Q(R). and let X = Spec(P).
We regard Rs as a subring of K for each x G X. Then Y{\J.@'x) = fiver ^v./"'' ""
arbitrary open set U of X.
Proof. (1) Consider first the case where U = D{f). Note that T{D{f).&x) =
R/. Since Rt C Pv for each .v 6 D(f). we know Rf C PUgcm ^- Let 0 be an
element of flxeou) ^- Then for each ,v G D(f). there exists /A G P — pv such that
£>(./\) C £>(/) and 0 = av//\ for some a, G R. Since £>(/) = Uvefl(/) D(/0 there
exists a finite open covering £>(/') = U"_, £>(/,) with /,■ = /X;. Here by putting
a, = aXi. we have a, = 0 • /', (1 < i < n). Meanwhile. /' = Yl','-i f>^'< ^or some
r > 0 and ^ G R. Let r = £;.'. , /W Then /Yfl = Y!L\ f>h<° = £"-i ^- = <'■
So 0 = c/f G P/, Therefore. f(D(/).^) = f\xeDU)R,.
(2) In the case of a general open set U. take an open covering U = (J/£/ D(ft)
(£)(/') 7^ 0 for each /) and look at a sequence
nu.ffx) -^Y[r(D{fi).ffx) =iYlr(D(fifj).^x).
iel i-i
Let (¢,),G; € n,G/ r(D(//).^A-) be such that &|D(/,/,) = C>|p(/',/,) for each /../'.
Then C,|D(/,/ ) and ¢,1/)(/,/ ) are equal as elements of A?/-,/ ;(C AT), where we note
that £>(/,/,) ^ 0. Hence, c, = ¢/ G DvgD(/',)nD(/,) ^ Applying this argument to
two arbitrary elements /./ of/, we know that £,■ = £; for each /\y G I. Let £ be
this element. Then £, G f]^,-L,Rx. Hence, by the definition of a sheaf. YXU.S'X) =
We shall consider the case where A! is a finitely generated algebra over a field k.
Lemma 3.18. Let R be a finitely generated algebra over afield k. let X = Spec(P).
and let M be the set of all closed points of X. Then M is a dense subset of X.
Proof. Let M be the closure of M. and let U = X - M. If U ^= 0. then there
exists a nonempty open set D(f) such that D{f) C U. Let m be a maximal ideal
of Rf. and let m = {a G R.a/\ G m}. Then k C P/m C Rf /m. where R/ /m is a

3. AFF1NE SCHEMES AND ALGEBRAIC VARIETIES
73
finite algebraic extension of k because R/ is a finitely generated Ac-algebra (cf. (1.1.15)
and (1.1.10(3))). Hence. R/m is a field and m is a maximal ideal of R. However,
m g £)(/) because f ¢. m. This is a contradiction. So M = X. □
Under the same assumptions as in the above lemma, we introduce the induced
topology on M from X. The induced topology on M restores the topology on X.
In fact, if/ is a radical ideal of R. then I = H/cm m by tne Nullstellensatz of Hilbert
(1.1.24). So I is restored by V(I) n M. We also have the following result.
Lemma 3.19. Let R be a finitely generated algegbra domain over afield k. and let
X = Spec(-R). Then T{U.@X) = Dyectw R^ wnere M is tne set of dosed points of
X.
Proof. Suppose Rf = Dv6/)(/')nw ^v f°r ^(/) - ^- Then Lemma 3.17 implies
nv.<9X)= [] «/= [] f| r,= fi /?v.
D(/)CC b(.f)QL .\(E»(/)nA/ \Gt'rw
Hence, we may assume [/ = £)(/) to prove the assertion. As the maximal ideals m
of 7? with f £ m correspond bijectively to the maximal ideals of Rf via m ^ mRf.
we have
n rx= p| /?m= n (/?/)„,/?,.
vGCnA/ /£m /"^m
So by replacing Rby Rf .we may assume / = 1. Now R C P)vew ^v- clearly. We shall
next show the opposite inclusion. Let 0 e Hvga/ ^v- anc^ let a = {a e 7?: a9 6 /?}.
Suppose ^ ¢ /?. Then a is a proper ideal of /?. Let m be a maximal ideal of R
containing a. Then 0 g Rm. Hence. fO e R for some / 6 /? — m. So / g a C m.
This is absurd. Thus. (\ga/ ^v = /?. □
We further make the stronger assumption that k is an algebraically closed field
and R is a finitely generated /c-algebra. Then we can write R — k[X\ X„]/I
as the residue ring of a polynomial ring over k. where I = (f\ /„,) as I is a
finitely generated ideal. Let m be a maximal ideal of R. Then R/m = k because
R/m is a finite algebraic extension of k and k is algebraically closed. Let m be the
pullback of m on k[X\ X,,]. Then m = m/I and m = (X\ — a\ X„ — a„).
where (a\ a„) € k" := k x • ■ • x k and /,(c*i a„) = 0 for each ;'. 1 < i < m.
n
Conversely, if (a\ a„) g k" satisfies /,-(a i a„) = 0 (for each i.\ < i < m).
then the maximal ideal rft = {X\ — a\ X„ — a„) contains I and m = m/I is a
maximal ideal of R. Then we say that (a\ a„) are the coordinates of a point m
(g M). Now we have the following.
Definition 3.20. Let k be a field and let A„ = k[X\ X„] be a polynomial ring
in n variables. We denote Spec(^„) by Aj? and call it the affine space of dimension n.
We assume, henceforth, that k is algebraically closed. Then the set of closed points
of A'l is identified with k". If R is a finitely generated A:-algebra, then R is given as
R = A„/I for some n > 0 and Spec(.ft) is identified with a closed set V{I) of A£.
Furthermore, the set M of closed points of Spec(^) is identified with V(I) C\k". We
call V(I) nk" an affine algebraic set defined by an ideal I of A,,. I its defining ideal.
and R = A„/I its coordinate ring. When we write X = Spec(/?). X(k) stands for
M = V(I)nk". We have V(I)nk" = V{y/7)r\k". and if X = X} U ■ • • U Xr is the

74
II. SCHhMhS AND ALGI-.BRAIC VARIF.TII-.S
irreducible decomposition, i.e.. \fl = pi n • • ■ n p,. is the prime divisor decomposition
and X, = V{p,) (1 < /' < r). then X(k) = X\{k) U ••• U X,.{k) is the irreducible
decomposition of X(k). If X is irreducible, we call X(k) an affine algebraic variety.
Then the coordinate ring R is an integral domain if we take / as a radical ideal I — \fl.
and its quotient field, which we denote by k(X). is called the rational function field
of X{k) (or X). We call an element of k{X) a rational function on X.
When M = X{k) is an afFine algebraic set. we regard the structure sheaf &x on
X as a sheaf tf^ on M in the following fashion: An open set V of M is written as
V = U n M. where U is an open set of X uniquely determined by V. We define
r([/n M.@m) = r(U.(9'x). Then <9M is a sheaf. We consider M as a local ringed
space {M.(fM).
As the above definitions show, local ringed spaces (X.(fx) and (M.^m)
determine each other uniquely. The difference between X and M is that X - M = {p G
Spec(.R);p ^ maximal ideal} as sets. For p Q X - M. V(p) = {p} is an irreducible
closed subset of X of dimension > 1. V(p) n M is an irreducible closed subset of M
of dimension equal to dim V(p). but the generic point p of V{p) does not belong to
V(p) n M. However. V(p) is restored from V(p) n M as the closure of V{p) n M in
A'. We call (X.&x) an ^'"f algebraic scheme and (M.@m) an affine algebraic set.
These are. of course, distinct objects. However, for the reasons explained just now.
we can and shall often treat them as the same.
For an affine scheme X — Spec{R). we denote by XKI\ an affine scheme Spec(^red)
determined by ^,ed := R/n. where n is the nilradical -\/lfi) of R. If X = Xm] as
schemes, we say that X is reduced.
We assume, for the moment, that k is algebraically closed. Let X C A", be an
irreducible and reduced, algebraic affine scheme, let R be its coordinate ring, and
let K := k{X) be its rational function field. Any element of R is represented by
an element of A„ = k[X\ X„\ If / e R is represented by F € A„. we write
/' = [F], For each .v G X(k). we set /(.v) := F(a\ a„) with the coordinate
expression .v = (ct\ a„). Then /(.v) g /c and this value f{x) is independent of
the choice of a representative F. Thus, an element / of R is looked upon as a A:-
valued function on X{k). If we express an element £, £ K as i^ = [F]/[G] (F. G G A„)
and G(<*| a„) ^ 0. then c(.v) := F{a\ a„)/G(a, a„) is determined
independently of the way of expressing l, — [F]/[G]. In fact, if c = [F']/[G'] and
G'{a\ a„) f 0. then FG' - F'G € I (:= the defining ideal of X). whence
F{a\ a„)/G{a\ a„) = F'(a\ a,,)/G'{a\ a„).
We then say that a rational function £ is regular at a point .v.
In terms of the regularity of a function. Lemma 3.19 is stated as follows.
Lkmma 3.21. In the above setting, we have
(1) For each x 6 X{k). R, = {c e K:c is regular at x}.
(2) For y e X - X{k). let Y = Jy}. Then R< = {c, e K:q is regular at some
point of YD X{k)}.
(3) Let U be an open set of X. Then r(U.&x) = {<? £ K'.£ « regular at every
point of U nX{k)}.
Proof. (1) Let £, e R^. Then c, = f/g with f.g e R and g ¢ px. Moreover.
/ = [F] and g = [G] with F.G & A„ and G ¢ «pv. where «p, is the pullback of p,
onto y4„. When we write .v = (at\ a„). G(a, a„) ^ 0. Namely. £, = [F]/[G]

1. AFK1NF. SCHEMES AND ALGEBRAIC VARIETIES
75
is regular at a-. Conversely, let c, e K be a function regular at x. Thus, we can write
i = [f]/[G] with G(e*i a„) ^ 0. Then G ¢^3,. Hence, if we put / = [F] and
g = [G]. then g ¢ p, := «p.v// and £ = //#. So £ e *,.
(2) Suppose £, is regular at a point x <= Y Ci X(k). Since pv D pi ■ Rr is a quotient
ring of Rx. By (1) above. £ G 7?v. Hence. £ G /?,.. Conversely, suppose £€/?,.
Write £, as c = //# with /. # e 7? and g ¢ p,. Let m be a maximal ideal of RK
containing \}VRX. Since R„ is a finitely generated /c-algebra. p := R n m is a maximal
ideal of ^ (cf. the proof of Lemma 3.18). Let x be a point of X(k) corresponding
to p. Then p,. C pv := p and g ¢ pv. Hence, a- e Y r\ X{k) and g(x) ^ 0. So. £ is
regular at .v.
(3) By Lemma 3.19. Y(U.(9X) = D.eunM R<- Hence. V(U.&X) consists of
elements of K which are regular at every point of U n M. □
Now coming back to Definition 3.3. we shall look into the properties of mor-
phisms of affine schemes.
Lemma 3.22. Let R and S be commutative rings, let X = Spec(7?). and let Y =
Spec(S'). Then we have
(1) A homomorphism of rings <p: R —» 5 gives rise to a morphism of local ringed
spaces (V <p): ( Y. ffY) -»' (X. &x).
(2) Conversely, given a morphism of local ringed spaces (/. 9): (Y.&y) ^> (X. tfx).
there exists a ring homomorphism <p: R —» S such that (f.0) = (atp.<p).
(3) Fix a morphism of local ringed spaces (f.0) ~ ("ip.ip): (Y.tfy) —» (X.@x).
If N is an S-module. f*(N) = (^]) ■ where Ny\ is the module N regarded as an
R-module via <p. Furthermore, if M is an R-module. then f*(M) = (M ®« S) .
Proof. (1) Let f = "<p. and let <p: &x —> j\&y be a homomorphism defined by
the natural ring homomorphism RK —> S ^y a/g" ^ ip(a)/ip(g)". where we note
that r{D{g).&x) = R, and r{D(g).f.0Y) = T(D(tp(g)).0Y) = S^k). For each
y € Y and x = /'(>')■ we have <p, : R, -+ (f^y), = lim Y(D(g). f,ffY) =
>\€/)(,s;)
lim S ( ). Since pv = </?~'(p,.) and g 4 px implies ip(g) 4 p,. we have a ring
>vE/)(,?) r
homomorphism lim Sirl,A —> Sv. This homomorphism composed with <2>. is
obviously a homomorphism of local rings (<£)*: R^ -+5,. So. ("</?• <£) is a morphism
of local ringed spaces.
(2) Let (f.0): (Y.tfy) —*■ (^-^V) be a morphism of local ringed spaces, and
let p> = 0(X): R —» S1. Then <^> is a ring homomorphism. We shall show that
(f.0) — ("<p.ip). For this we shall make use of the following commutative diagram
0x(X)^+0y(Y)
#x.x ^(/^k)v = Hm <?r(f-](U)) -jl^*y,
vet'
where U ranges over the set of open neighborhoods of x. If x = f(y). then y e
f~x(U); whence, there exists a natural ring homomorphism /; as given in the above
diagram. We have p ■ 0y = (0#)y. Namely. /; • 0X is a homomorphism of local rings.

76 11. SCHEMES AND ALGhBRAlC VARIETIES
Hence (p~ ' (p,.) — px. So. / = "ip. Moreover, comparing the definitions of (tp)* and
[0#)y. we know that {<p)# = (0#)v for each y e Y. Hence (£)# = ()#. So. <p = 0.
(3) For an open set D(g) of*. r{D{g).f,(N)) = r{D{<p(g)). N) = N[v] ®R Rv.
If D(h) C £)(#). the restriction morphism/;/,,, := pn(i,)D(i;)'- F{D{g).f*{N)) —>
r{D{h).f* (N)) coincides with the restriction morphism lN ®r /;/,,,: A^j ®« /?? —>
tfM&« «,,. Hence. f*(N) = {NM)".
For an ^-module M. define a homomorphism of ^-modules j: M —> (A/ (¾
5)[^] by «) h m g 1 which gives rise to a homomorphism of ^-Modules /: A/ —»
/,((MK/;S)~) and a homomorphism of ^-Modules (j-)8: /*(M) -> (A/ ¢¢/} 5)"\
We shall show that (/)# is an isomorphism. For each y g Y. we have
f*{M)y = [f*M ®f,6x &y)y = M, ¢0^ x &y.y = (M ®R 5) 8,v 5,.
where .v = /(>>). This implies that (j)\ is a homorphism of 5r-modules and the
identity morphism when restricted on M ¢0 l.s. So. (/), is an isomorphism for each
y e Y. Therefore. /*(M) ^ (M ^k 5)\ □
Theorem 3.23. With the same assumptions as in Lemma 3.22. the following
assertions hold:
(1) For an exact sequence of scattered S'y-Modules
0-^ -^ ^2 -^ 5^ -> 0.
?/k' sequence
0 -> ./,5H ^ /*^2 ^ /,5^ -+ 0
/.v an exact sequence of scattered &x~ Modules.
(2) If 5 is a noetherian ring, we have R' f \!F = (0) {all i > 0) for every
quasicoherent @y-Module &.
Proof. (1) It suffices to show that the morphism of stalks (/*/?).v is surjective for
each a- £ X. Let £, e (/*^)v- Then there exist an open neighborhood U of .v and .v 6
V(U. /,^,) suchthat£ = .vv. Here Y{f~\U).p): Y{f-l{U).^2)^Y{f-]{U).^)
is surjective by (1.2.19). Choose t e r(/-' (U).^2) such that ,v = Y(f~]{U).p){t).
and let /7 = tx. Then £ = (/'*/?) v (/7). So (/*/?) v is surjective. It is clear that if ^" is
a scattered ^> -Module then /*■?" is a scattered ^--Module.
(2) R1 ftS? coincides with the cohomology group defined by a scattered
resolution of W. In fact., by making use of the assertion (l) above and the argument in
a proof of problem 1.2.5 (see the hint), we know, first of all. that the cohomology
groups defined by a scattered resolution of & are independent of the choice of a
scattered resolution of 9~. We denote these cohomology groups by R1 f \&'. Then
{/?'/*(*)}/>() is a cohomology functor. Here we have to extend the definition of
cohomology functor (see Chapter 2 of Part I) as follows. Namely, a functor assigning
to an $y-Module & a collection of ^-Modules {K' (SO }/>o is a cohomology functor
if the following conditions are satisfied:
(i) K'\^) - f*&~.
(ii) K'(jf) = (0) (for each i > 0) for an injective ^-Module JF.

.1. AFFINh SC'HFMF.S AND ALGEBRAIC VARIETIES 77
(iii) For an exact sequence of ify-Modules
0 _» grx il jr2 i> jr, _> o.
there is the following cohomology exact sequence
^ k1 (^-,) ^ k2^,) ^ ► k"{Fi) ^ /r (.%)
^^"(^)^^""(^,) -^---.
If <? is an injective <f>-Module. then J^ is a scattered sheaf (cf. (1.2.19)). Then
R'f*<f = (O) for each i > 0 by (1) above. For an exact sequence of @Y-Modules
the standard scattered resolutions of ^ (j = 1.2.3) (cf. Problem 1.2.5) yield by a
straightforward argument a cohomology exact sequence
> R"f*F{ ^ R"J\Sr2 £ R"f\^ ^ R"^f^i -> ■ ■ •
We conclude that R' f ^ = R'f^ (for each i > 0).
Now let J be a quasicoherent (9Y -Module. Write SF = N. where N is an 5-
module. Let
be an injective resolution of an S-module N. Then
is an exact sequence of 7?-modules. Hence, we obtain an exact sequence of
quasicoherent ^--Modules
u->./*./ -* j *j -> ./*,/ —>■■■ ^> j *jt —> • ■ •
where /" = /". If we note that
o _»jr 1» /° ^ ^/1 '1\ , f ih ...
is a scattered resolution of & (cf. (3.13)). we have R' f\SF = (0) (for each ;' > 0).
So /?''/*.y = (0) for each i > 0. ' □
Let /c be an algebraically closed field, and let X(k) and Y(k) be respectively the
affine algebraic sets over k with the coordinate rings R and S. A mapping of sets
f : X{k) —> Y{k) is called a movphism of affine algeraic sets if there exists a /c-algebra
homomorphism <p; R —> S such that the following diagram is commutative:
Spec(S)-^Spec(7?)
Y(k) '-^X{k).
In this definition, <p is not necessarily determined uniquely by / . However, if X{k)

7X
II. S( HhMhS AND ALCihBRMC VARIEI1LS
and Y{k) are affine algebraic varieties. <p is determined uniquely by / as shown in
the next lemma.
Lemma 3.24. Let X{k) and Y{k) be affine algebraic varieties defined over an
algebraically closed field k. and let f: Y(k) —> X(k) be a mapping of sets. Then
the following conditions are equivalent:
(1) f is a morphism of affine algebraic varieties.
(2) (i) f is a continuous mapping.
(ii)" Ifg e k{X) is regular over X{k). i.e.. g e R : = Y{X.@X). then f*{g)
is regular over Y{k). i.e.. f*{g) £ S : = Y{Y.@Y). where f*{g){y) =
g{f{y)) for each y e Y.
The homomorphism of coordinate rings f. R —> S {see the above definition) is nothing
but f*.
Proof. (1) implies (2). By definition, there exists a &-algebra homomorphism
tp: R —> S such that / = 'Vlr(A)- Obviously, f is continuous. For g £ R and
y £ Y(k). f*(g)(y) =g(f(y)) = g("<p(y)) = <p(g)(y). Hence, f*(g) = <p(g) 6 S
as a function on Y{k).
(2) implies (1). Express R as R = k[X\ X„]/I with I = J7. Let £,■ be the
residue class of Z/ (mod/), and let ¢: k[X\ X„] —> S be a ^-homomorphism
such that ¢(^,) = ./"*(£/) for each /. For G e /. we have
*(g)(>-) = G(r(so /*(£„))(>■) = G(.r(«.)(j) /*(<;„)(>■))
= G(C, (/(>')) C,(/(>•)))■
where £/(/00) = «; if /0') = (<*i «„) £ k". Since /(>>) e Af(/c) and G e J. we
have ¢(¢7)00 = G{a\ a„) = 0. Hence. ¢ induces a ^-algebra homomorphism
9?: R —> S. For a maximal ideal p, of 5. c^~'(pr) is a maximal ideal of R which
corresponds to a point x = {a\ a',). Writing /(>') = (a) a„). we have
a,' = &(.v) = £,(VO0) = <p(ti)(y) = W)00 = ./^000 = «/•
Hence, it follows that / = 'VI y (/„-)■ Suppose f is induced by two ^-algebra ho-
momorphisms ip.ip': R —> S. Then for each _y- £ ^(^) and g e R. we have
¥>(s)00 = /*(*)(>•) = ^0,-)00- So ^00 - </(g) e aer(/,)P, = V^0) = (0)-
Namely. ip{g) = ip'{g) for each gGR.lt follows that ip = ip'. D
Here we give some explanations of the notation X{k). Let R be a finitely
generated algebra over an algebraically closed field k. and let X = Spec(i?). For a
closed point x of X. R/p-, = k. The residue homomorphism R —> k gives rise to
a morphism of affine schemes Spec(Ar) —> Spec(/?). Note that Spec(Ar) consists of a
single point and the structure sheaf is k. Conversely, given a morphism of algebraic
sets over k. f: Spec(fc) —> Spec(/?). it is given by a ^-homomorphism ip: R —> &.
Kery? is a maximal ideal of i? and corresponds to a point /(Spec(/:)). This point
is determined independently of the choice of <p. Therefore, we have a bijection
/: Spec(£) —> X = Spec(/?); a morphism of
"gebraic sets
We denote the last set by Horn/, (Spec{k). X) (or X{k) in a more abbreviated
notations).
X{k)

3. AFF1NE SC"HEMES AND ALGEBRAIC" VARIETIES
79
II.3. Problems
1. Construct an example of an affine scheme X — Spec(/?) for which X is a
noetherian space but R is not a noetherian ring. One way of constructing such
an example is the following. Let A be a commutative ring, and let M be an
^-module. In the direct sum R = A >]) M we define a ring structure by (ti\. x\) +
(tf2..\"2) = («i +«2..vi + a"2) and («i.A"i) • («2.a'2) = («i«2-«2A'i +C/1A2). The
identity element is (1. 0). and elements .v of M are identified with (0. a). Hence
(O.a-) • (O.y) = (0.0). Namely. M2 = (0). Hence Spec(/C) = Spec(A). We call
R the idealization of an ^-module M. Here we take a noetherian ring A and an
^-module M which is not finitely generated. Then R is not noetherian.
2. Let A be a noetherian domain, and let {p;.};.eA be the set of prime ideals of A
such that htp;. = /• for each /, e A. Let A^p,) = Q{A/p;_) and I ~ >.D;.eA^(p;.).
Then show that for a quasicoherent Module I over X = Spec(^). we have
Supp(/) = U;eA ^(P;.)- So Supp(/) is not a closed set of X provided A is
an infinite set.
3. Verify that for an exact sequence of /("-modules
0^ L^ M h N ^0
and an /("-module P. the sequence
L ®R P "^'' M ®r P ^ N 9:R P -> 0
is an exact sequence. (It is easy to show that ft c$ \F is surjective. In order to
show that Ker(/7 ® l/>) = Im(a (¾ \F) one needs to go back to the definition of
a tensor product M <x>« P.)

CHAPTER 4
Schemes and Algebraic Varieties
In order to proceed further, we need the concept of a scheme which is more
general than that of an affine scheme.
Definition 4.1. Let (X. &x) be a local ringed space. (X.&x) >s called a scheme if
for every point x of X there exist an open neighborhood U of x and a commutative
ring/4x such that (U.@x\u) is isomorphic to (Spec{Ax).Ax) as local ringed spaces.
An open set U of a scheme (X.&x) is called an affine open set if [U.@x\u) ls
isomorphic to an affine scheme (Spec(A).A) as local ringed spaces. For schemes
(X.@x) and (Y.&y). a morphism of local ringed spaces {f.<p)\ {Y.&y) —> (X.@x)
is called a morphism of schemes. As an abbreviation of (/. <p). we frequently write
/: K —> X. If there exists such a morphism /: Y —» A', we call F a scheme over A'
and call / the structure morphism. If /: F —+ X and g: Z —> A" are schemes over
X. we define
Homj-( Y. Z) = (¾ : Y —* Z. a morphism of schemes with f = g • h}.
In the next lemma, we list some of the elementary properties of schemes.
Lemma 4.2. Let (X.&x) be a scheme.
(1) An affine scheme is a scheme.
(2) The set of all affine open sets of [X.&x) is an open basis of the topology of
X.
(3) (X.@x) is a To-spaee. but it is not necessarily a T\-space.
(4) An irreducible closed set F of X has a unique generic point. Namely, there
exists a unique point x 6 F with F = {x}. If F fl U ^= % for an open set U of X.
then x £ U.
(5) Let JF be a quasi-coherent Ideal of' @x- i-C-- *? is a sheaf of ideals of&x which
is a quasi-coherent <fx-Module, let Y := Supp^/J7). and let <fy ■= @xl^■ Then
[Y.&y) is a scheme. {The scheme (Y.&y) expressed in this manner is called a closed
subscheme of X. If a scheme (Z. @z) is isomorphic to a closed subscheme [Y.&y) of X
via a morphism a: (Z.&z) ~* [Y.&y). then the morphism ((7.0): (Z.&z) -* (X.&x)
is called a closed immersion or a closed embedding, where 0: &x\y ^&x /<? "-* &z ■)
(6) For x e X. Spec(^V.Y) = {y 6 X\x e {y}}. {For two points x.y of X. if
x £ {y}. we say that x is a specialization of y and y is a generalization of x.)
(7) For an open set U of X. (U.&x\v) is a scheme. {A morphism of schemes
((7.9): (Z.&z) —+ (X.&x) is called an open immersion if a(Z) is an open set of X.
a induces a homeomorphism between Z and o(Z). and 0#: a*&x —► &z induces an
isomorphism between &x\n(z) and &z.)
81

82
II. SClllvMI S AND ALGI.BRAIC VARIETIES
Proof. (1) and (2) are obvious.
(3) Let v. v be two points of X. and let U be an affine open neighborhood of
.v. Suppose y £ U. Since .v and y are two points of an affine scheme, the remark
given before Definition 3.3 implies that there exists an open set containing one of
x.y and not containing the other. Hence, the topology of X is a 7~b-topology.
(4) If U is an open set of X with U n F ^0. then there exists an affine open
set V of X such that V C V and fnF^l Since F = ~{V C\F) U(Fn(I- V))
and F is irreducible. F = (V n F). Namely. VC\F is dense in F. Suppose VnF =
(V n F,) U (V n /¾). Knfi^Kn F. and rnf2^ fnf for closed sets F\. F2 of
X. Then we have
F = (FnFi)u(FnF2)u(fn(i- k)).
F n F, g F. F n F2 g F
which contradicts the irreducibihty of F. Hence. V n F is an irreducible closed subset
of P. By Lemma 3.7. there exists a point a <E V n F such that fnf = {a} in
F. Hence. F = {a} U (F n (X - K)). So by the irreducibihty of F. F = {a}. If
F = Jx7}. then a' e V n F. So by Lemma 3.7. a = a'.
(5) Since Sx/J^ is finitely generated. K := Supp(<fv/J^) is a closed subset of X.
For each v £ K. let U = Spec(^) be an affine open neighborhood of v. Then there
exists an ideal I of A such that S\L- = I. Y r\U - V{I). and &YnL
= Spec(^//).
Hence. Y n t/ is an affine open neighborhood of a in K. Thereby. F is a scheme.
(6) Let V = Spec(^) be an affine open neighborhood of .v. Then @x.\ — ^v (the
localization of A by a prime ideal pv). Suppose y £ {v}. Then y 6 V and pv D p,
in A. So j- £ Spec(/4.J. The converse also holds. So Spec(^v) = {y e X:x £ {>•}}.
(7) For each v £ U. we can choose its affine open neighborhood V so that
V C £/. Then (£/.^|t ) is clearly a scheme. □
Define the sheaf of nilradicals JV by
U ^yT(U) = the nilradical ofr{U.&x).
Then jV< is the nilradical of &x.v f°r eacn v £ -^- Moreover. >" is a quasicoherent
Ideal oi&x- Hence. {X.@xjjV) is a closed subscheme and Supp(@x/J^) = X. We
denote this closed subscheme by Xrej and call it the reduced form of X. (Compare
the definition with the one given before Lemma 3.21.) We say that X is reduced if
If Y is a closed subset of X. there is a unique reduced scheme structure on Y.
In fact, let f/ = Spec(^) be an arbitrary affine open set of X. Then Y HU — V(I)
with I = \/7. For another affine open set Vj= Spec(B). write V C\ Y = V{J) with
J = \fj. Then /|rnr = ^|rni • (Compare /v and Jx for each x £ f/ n I7.) Hence,
there exists a quasicoherent Ideal .y such that Jf\i = I. It is clear from the above
construction that Y = Supp^^/J7) and {Y.&xj.J) is a reduced scheme. Suppose
Y --= Supp(d?jr/,/). ,/|r = J. and J = sp. Then f/nf= F(/) = F(7); whence.
I = J. Hence. ,y = ,/'. So there is a unique reduced scheme structure on Y. We
also denote it by yrecj.
If X{) is a connected component of X. then A^ is also an open set of X. Hence.
{X[).0x\x„) is a scheme. If X — TJ,e/ X, is the decomposition of X into connected
components then X is regarded as a disjoint union of schemes {(Xi-@x\x,)}i£i-
Conversely, given a set of schemes {(Xi.0Xl)}iei- we define on a direct sum of topological

4. SCHEMHS AND ALGEBRAIC VARIETIES
83
spaces X = T].e/ X, a sheaf ^¾ by @x\x, =@x,- Then [X .&x) isascheme. We call this
scheme a direct sum of {(A";-^,))/6/ and denote it by ]J;e/ A',-. It is straightforward
to verify that the direct sum satisfies the following property:
Horn I JJX-,. T ] =Y[Hom(A/ T) for every scheme T.
ve/ J /el
where ]T,e/ stands for a direct product of sets and Hom(-. •) is the set of morphisms
of schemes.
We shall prove the following lemma concerning local properties of schemes and
morphisms of schemes.
Lemma 4.3. (1) Let {Wj}iei be a set of schemes satisfying the following two
conditions:
(i) For each i.j e I. there exist open sets £/,-/ of Wj and £//,- of IV,- and an
isomorphism of schemes 0,-,-: {U,-,.@w\Vi) —> [V-,-,.&w, \u„) such that 0,, •
6jj = 1 and On = 1- where 1 stands for the identity morphism.
(ii) For each i. j.k £ I. OjfV-,-, n Vik) = U,, n Ujk and 0kj ■ 0,i = 0ki on
£/,7 n £/,-,.
Then there exist a scheme W and the open immersions £j: Wj —+ W for each i £ I
such that W = [J/e/ £/,- with U-, = £i{Wj) and £, induces an isomorphism between 11-,-,
and Uj n Uj to the effect that £i\L\, = (£/1(/,,) ■ Oj-, for each i.j 6 I.
(2) Let X and Y be schemes and let X = (J,- ; U-, be an open covering of X.
Suppose we are given morphisms of schemes f,-: £/,- —> Y (i £ I) such that .//lent, =
//1 (/, ru -, for each i.j £ I. Then there exists a morphism of schemes f : X —> Y such
that fj = f IL; for each i £ I.
Proof. (1) First of all. there exist a topological space W. its open covering
{Uj}iei and homeomorphisms £,-: W-, —> £/,- (/ e I) such that £,(£/,7) = £/,- n £//
and (//,- = (£j\cn)~l • (ii\f ) for each i.j £ /. We define a sheaf of commutative
rings (fV on W as follows: Let U be an open set of W. If E/ C £/- (for some r)
then T{U.0w) = {.v • £r';.v £ r(£rl(U).0Wi)}. If E/ C E/, n £//. then .v . £rx =
(.v • 0,-,-) • £~ . where .v h^ ,v • 0-,-, is an isomorphism between T{£j {U).@w,) and
r(^/"'(C/)• <%,)■ This implies that the definition of T{U.@W) is independent of the
choice of i e / with E/ C £/,. So we can define the stalk &w_x for each x £ H7
and introduce a topology on &w = LLg^^Vv such that the natural projection
71: &w —► W is a local homeomorphism. We then define &w as the sheaf of local
sections of n (cf. (1.2.5)).
It is clear from the above construction that £,-,: {W-,.@w) —+ {Uj.@w\i,) is an
isomorphism of local ringed spaces. Hence. W is a scheme and £,: W, ^+ £/,■ <-» H7
is an open immersion.
(2) Patching {/,-},-e/ together, we can define a mapping of topological spaces
f: X —* Y. We can also define a homomorphism of sheaves of commutative rings
<p#: f*@y —> &x by patching together homomorphisms ipf: f*@Y —> ^r, associated
with /,-. In fact, we have only to note that f*/fy\u, = f*^r- and f*@Y\u = f*j@Y\v
and <£>f |[- = <p*\u provided U C £/,- n £//. Then / is a morphism of schemes such
that/, =/| [-,. □
The following property of schemes is also important.

84
II. SCHEMES AND ALGEBRAIC VARIETIES
Lemma 4.4. Let {X.@x) he a scheme, let {S.@s) = {Spec(R).R) be an affine
.scheme, and let (f.<p): {X.<fx) —> (S.tfs) be a morphism of schemes. Let A =
T(X.<9x). and let a = T{S.ip): R —> A. Then there exists a unique morphism of
schemes {g.y/): {X.@x) —» (Spec(^).^) such that {f.<p) = ("v. a) ■ {g.y/).
Proof. For an affine open set U = Specie) of X. the restriction morphism
Pcx- A = Y{X.@X) —► Ai = T{\J.&x) gives rise to a morphism of affine schemes
ge '• {U.<fx\i) —>■ {Spec{A).A). We shall show that ge\cnc = ge'lenc f°r another
affine open set U'. For this, it suffices to show that ge\y = gy for any affine
open set V = Spec(2?) contained in U n U'. Meanwhile, the restriction morphism
pye' Ai —» B = r(V.ffx) induces a morphism of schemes "(pye)'- V —* C/.
which turns out to be the natural open immersion K <-» C/. Hence, the relation
Pve • Pcx = Pvx implies gc\r = gy. Since X has an affine open covering, i.e.. a
covering consisting of affine open sets, we know by Lemma 4.3(2) that there exists a
morphism of schemes g: X —» Spec (,4). Let f: X —» S be a morphism of schemes
as in the statement. The restriction f\e'- U —> S of / onto an affine open set U
corresponds to a ring homomorphism aL-: R—^Ae (Lemma 3.21). For V C U C\V
as above, /d corresponds to /;jr • ft/ = cr = Pec • ctl'- Since ^¾ is a sheaf, this
gives pence • ffr = Pence • <?e'- Hence, by the definition of sheaf again, we
know that there exists a ring homomorphism a: R -^ A such that 07; = /)(.■x • f-
(More precisely, consider an affine open covering {U,};eA of X and apply the above
argument.) Hence. f\e decomposes as
U ^ SpecU) % S.
Since U is an arbitrary affine open set. we have f = "a • g. The uniqueness of g
should be clear by the above construction of g. □
Let S be a scheme, and let f: X —» S and g: Y —» S be schemes over 5. A
triple {Z.p. q) consisting of a scheme h: Z —> S over S and morphisms of S-schemes
p: Z -^ X and </: Z —> Y is called & fiber product of Z and K over S if the following
conditions are satisfied:
For every S-scheme T. a mapping of sets
Horns(T.Z) -> Homs(T. X) x Hom.s(r. Y). u^(p-u.q- u)
is a bijection.
We denote Zbylxj Y as well. If Zxj Y exists, then morphisms of S-schemes
(S-morphisms. for short) a: T —> X and fl: T —> F" determine an S-morphism
7" —> A' x,v K. which we denote by (a./?).v or (a./?). If Z exists, it is determined
uniquely up to an S-isomorphism. More precisely, if an S-scheme h'\ Z' —+ S
together with S-morphisms p': Z' —+ X and </': Z' —> Y satisfies the same conditions
as the triple {Z.p. q) does, then there exists an ^-isomorphism 0: Z ^+ Z' such that
/> = // • # and q = q' • 0. This isomorphism 0 is uniquely determined.
Theorem 4.5. A fiber product X xs Y of S-.schemes {X. f) and (Y.g) exists.
Proof. We shall prove the assertion in several steps.
Claim 1. Assume that a triple {Z.p.q) consisting of an .S-scheme Z and S-
morphisms p: Z —> X and q: Z —+ Y gives a fiber product X x$ Y. Let U and
V be open sets of X and Y. respectively, and let W = p~]{U) n (/''(I7). Then
IF = C/ x s- K. where C/ and V are viewed as S-schemes via f\e and g| y. respectively.

4. SCHEMES AND ALGHBRA1C VARlLTlhS
85
Proof. Given S-morphisms v. T —> U and w. T —> V with /|f • v = g\r -w,
there exists a unique S-morphism u: T —> Z such that ?> = p • u and w = q • u.
Then the morphism w passes through W7 as w: T —> W7 ^ Z. Conversely, if an
S-morphism u: T —» Z passes through W7. then /; • u and </ • u pass through £/ and
V. respectively. D
Claim 2. Let Z be an S-scheme. and let p: Z —> A" and q: Z ^ Y be S-
morphisms. Let il = {£/,■},■ ey and 2J = {V/jjeJ be open coverings of X and Y.
respectively. Let Wr, = p~x(U;)r\q~\Vj) for each (i.j) e / x J. If U), = [/, x.s- K,
for each (/. y) e / x J. then Z = X xs Y.
Proof. We show that
Hom.v(r.Z) ->Homs(7\ A") x Hom.s-(7\ Y). u^(p-u.q-u)
is a bijection.
Injectivity. Suppose we are given S-morphisms u.u': T —> Z such that p • u =
p • u' and q • u = q • u'. Since
M-'(^7) = w-|/7~'(^)nw~1^-'(i//) = ^)^(^)0(^)-^^-)
and since W7,, = E/, xs Vj, we have w|„-i(fr j = u'\u,-\^w y Obviously.
{u~^{Wjj); (/./) £ / x /} is an open covering of 7\ whence w = w'.
Surjectivity. Let u: 71 —> X and w;: 71 —+ Y be S-morphisms. Set 7),- = w ~' (U,■) n
iu_1(Vy). Then there exists an S-morphism u,,-: 7),- —> W7,-,- such that p • w,,- = v\Ti
■xaAq-Ujj = Hr„- If 7), n 7)., ^ 0. then u,, (7), n 7),,) C/,-'(£/,,) rig"1 (^/) = Wk,'.
Hence. u,,(7), n 7),,) C Wr, n W7,,,. We also have «/,,(7,, n 7,,,) C Wn f~l W7,,,.
On the other hand, since W7,-, n Wk, = /?~'(E/, n £4) n q~\Vj n F,). we have
W7,-, n W4, = (£/,- n £4) x.s- ( F)- n K,) by (1) above. Furthermore, since/; • w,-/|7-,,07^ =
v|y„nn, = /? • Uki\T,,nn, and <? • w,-,|r,,nr,,, = w|r,-,nr,,, = <7 ■ "/>/|r,,n7v we have
Ui/\r,,nTu = "A/|r,,n/,,, by the definition of fiber product. Noting that T = (J, 7),-,
we then know that there exists an S-morphism u: T —> Z such that «,, = m)^ for
each (i. j) € I x J. □
Claim 3. Let it = {£//},E/ and 2J = {K,-},-ey be open coverings of X and K.
respectively. If £/,- xs Vj exists for each (i.j) £ I x J. then X xs Y exists.
Proof. Put a = (i.j), briefly. Let (Za.pn.qa) denote a fiber product £/,- xs- F,
and its projections. Let /J = (k. 1). By (1), Za/, = p"1 (£/, n £//,) n &71 (K, n K,) and
Zpa = Ppl(Uj n £//,) n^'C7, n K,) are fiber products of £/,- n £/,, and F, n K,. and
by the uniqueness of a fiber product, there exists an S-isomorphism Onp: Zp„ —» Zap
such that /;„ • Oap = pp and </a • Oap = qp. Moreover, since Zya n Z-.p is a fiber
product of E/, n Uk n £/,„ and I7, n F, n K„ for y = (m.n) e I x I. we have
6ap • Qpy = ftv; on Zya n Z;,,;. Hence, by applying (1) of Lemma 4.3 to {Za(a e
7 x J).Zap (a. ft £ I x J)}, we know that there exist an S-scheme Z. an open
covering {Wn}nelxj of Z, and S-isomorphisms £0 :Z„^ Wa(ae/xJ) such that
£,a(Zltp) = Wan Wp = £p(Zpa) and 0a/J = (6»|z,l/()_1 ■ (^|z/(J. Here, since
/?« • (^rt I»'„nw„) = Pp- (<ip Iw„nir/()
i if,,nicj - <lp • (c,p I w„nw„)-

86
11. SC'HF.MFS AND ALGEBRAIC VARIETIES
we know by (2) of Lemma 4.3 that there exist 5-morphisms p: Z —> X and q: Z —» Y
such that p\wn = pa • £,71 and q\w„ = <?« • £,7'• Then we have only to show that a
triple (Z.p. q) is a fiber product of X and y over 5. We will be done by (2) above if
we show that Za is 5-isomorphic to p_'(£/,-) n q~{(V,-) for each a = (/./) e I x J.
For /? = (/:. /). we have
/>-'(£/,-)ntf-'tjgn (*> =^^^^)0^-^,0 v,)
= tt)(pp\virMJk)c\q-\vjr\v,))
= £p{zfia)= w„cmvp.
Since /? £ / x / is arbitrary, we have /;_' (t/,■) n q ~' (Vt-) = Wa. which is 5-isomorphic
to Z„. Hence. Z is a fiber product of X and Y over 5. □
Claim 4. Let {5,1,(=/ be an open covering of 5. let X, — /-1 (5,-). and let
r,- = ^-1 (5,-). If a fiber product X, xs, Y, of {X,-. f\Xi) and (K,.g|r,) exists for
each i e /. then there exists a fiber product X xs y of {X. f) and (y. #).
Proof. Since {A",-},-e/ and {^/}/e/ are open coverings of X and y. respectively,
we have only to show by (3) above that a fiber product X-, xs Y,- exists for each
(i.j) £ I x J. First of all. X,- xSi Y,- is a fiber product of X, and Y, over 5. where X,
and y,- are regarded as 5-schemes via 5,■ ^-> 5. Let A",-,- = X, n A',- and y, = y n y;.
By (1) above, there exists a fiber product X-,-, xs y,; which we denote by Z,;. We
can show, by verifying the conditions in the definition of fiber product, that Z,; is
5-isomorphic to X, xs Yr So X, xs Y) exists. By (3) above, a fiber product X xs Y
thereby exists. □
Let {5,},e/ be an affine open covering of 5. By (4). it suffices to show that
X; xs- Yj exists. So we may assume that 5 is an affine scheme. Now let {£/,},G/
and { Vj}j£j be affine open coverings of X and Y, respectively. By (3). it suffices to
show that Uj xs Vt exists. So we may assume that X and Y are affine schemes as
well. Namely, suppose /: X —> 5 and g: Y —> S are given by ring homomorphisms
ip: C —> A and y/\ C —> 5. respectively. With /4 and 5 viewed as C -algebras by ip and
y/. we show that Z xs- Y = Spec(^ ®c B). Let T be an arbitrary 5-scheme. Note
that every 5-morphism from T to an affine scheme is factorized by SpecT{T.@T)
(Lemma 4.4). Hence, in order to show
Hom.s(T.Z) = Hom.s-(r.Jr) x Homs(r. Y).
where
Z = Spec(^ ®r B).
we may assume that T is also an affine scheme. Let T = Spec(R). Then we have
only to show that a mapping between the sets of C-algebra homomorphisms
Homc (A. R) x Hom<--(fl..R) -> Homc(^ ®c B.R)
{a.p) ^ I a <8/? : 5^ a,-® 6,- >-► J]a(a,■)£(&,-) )
is a bijection. This is a well-known result. The proof is left to the reader. □
The underlying set of X xs Y is not necessarily identical with the product of
underlying sets {(.v. y):.\ e X.y e Y} (problem II.4.1). In the last step in the proof of

4. SCHEMES AND ALGEBRAIC VARIETIES
S7
Theorem 4.5. we saw that the fiber product X xs Y of affine schemes corresponds to
the tensor product A &>(- B. So properties which are common to the tensor products
are inherited by the fiber products of schemes as shown in the next lemma.
Lemma 4.6. Let (X. f) and {Y.g) be S-schemes. Then the following assertions
hold:
(1) For (X.f) and (S. l.v). we have X x.s S = S xs X = X.
(2) Let (S'. it) he an S-.scheme, and let Xs> = X xs S'. Let fs, : Xs> -> 5" and
u x '■ Xs> —> X he the natural projections. Then u • fs* = f -ux- (We call fs1 : Xs> —> S'
the base change of /': X —+ S bv u. We also write fs> = f x.s S'.)
(3) Xs, xs, y;s-,' = (X X,v Y)s>.
(4) f~l(U) = X x.v U for an open set U of S.
Each assertion is easy to prove, so we omit the proofs. Base changes which
we frequently have to consider include the following four kinds. In the following.
f: X —> Y signifies a morphism of schemes.
(1) Let y be a point of Y. and let k(y) be the residue field of a local ring <9;-.,..
If V = Spec(B) is an affine open neighborhood of y. then the point y corresponds
to a prime ideal q of B and k(y) = Q(B/c\) = Bq/qBq. Hence, the composite of the
natural ring homomorphisms B —> B/q <^-> k(y) gives rise to a morphism of schemes
iy: Spec(k(y)) -+ V <-* Y. The base change of/ by /,. is denoted by /,: Xv —>
Spec(Ar(j)). where Xv = X x Y Spec(k(y)) and /, = / x r Spec(/: (>>))■ This definition
of (Xv.fr) is determined independently of the choice of an affine open neighborhood
V. We call (Xr. fy) the fiber of / over y. In order to look into the structure of Xv.
consider the inverse image f~x(y) as a set and consider a family of affine open sets
{L/,},^ of X so that U,eA V;. b /-'(>')• Then X, = U/£A U>. x r Spec(A: (_>■)) and
£/, x v Spec(Ar(j)) = Spec(^;. <g>B k(y)). where t/;. = Spec(A;).
(2) Let (& .tff) be a discrete valuation ring. Consider the case where Y =
Spec(<f). Then Y consists of two points n and .v. where n corresponds to the ideal
(0) of & and s to the maximal ideal t& of &. Hence. /: X —> K consists of two
fibers A^ and Xs. where Xn is defined over the field K = Q(0). i.e.. Xn is a scheme
over Spec(AT). and Xs is defined over the field k : = &/t&. We say that Xs is the
reduction of X,r In general, for a scheme 7 and its point y such that the local ring
&y.\- is ar> integral domain, we can find, under some additional conditions, a discrete
valuation ring (&.t&) such that &Yx < & and Q(&Y.X) = Q(@) (cf. Lemma 4.12).
If we put A> = X x Y Spec(<f) and' ffi = f x Y Spec(<f). then fff : Xff -> Spec(<f)
fits in the situation considered above. We call Xs a specialization of A/ We use this
situation to observe the extent to which the properties of X,t are inherited by Xs.
(3) Let U be an open set of Y. let XL = f ~](U). and let // = / x Y U. Then
//: A/ —^ C/ is the restriction of / onto U. We have the next result.
Lemma 4.7. Lef ^ be a quasicoherent $x-Module. and let ,?/ be the restriction
of & onto an open set f~l(U). Then (R' f^)L- =1 R'(f L)^v for each i > 0.
Proof. An injective ^-Module .J is a scattered sheaf (1.2.19). Hence. J/ is
a scattered <f/ -Module. Thus, if
o->^^° a^i a;..._^« a;...
is an injective resolution of ^.

88
II. SCHEMES AND ALGbBRAlC VARIETIES
is a scattered resolution of SFL. On the other hand, note that [f\9r)i = ifc)*^
and {j\.y)[j = (/[/)*,yL■. Then the above two complexes yield the isomorphisms
as stated. Here we note also that R'{fu)*&~L is obtained by a scattered resolution
of 5s/ (cf. Proof of Theorem 3.23). □
(4) We say that a morphism of schemes v: Z —> Y is aflat morphism if &Zz is a flat
<^V., -module for each z £ Z and v = r(r). The base change f z : Xz := X xY Z —+ Z
by v behaves in terms of cohomologies in the same fashion as fL in Lemma 4.7.
Namely, we shall state the following result without giving a proof.
Lemma 4.7.1. Let SF be a quasicoherent 0x-Module. and let !FZ be the inverse
image of & on Xz. Then (R1 f^)z = R'{fz)*^z far each i > 0.
Now let f: X -* S be a morphism of schemes. Define the diagonal morphism
Ax/s ■ X —> X x.v X as the morphism whose composites with two projections from
X xs- X onto X are the identity morphism of X. As a mapping of sets. AX/s has
image Ax/s{X) = {(x.x):x e X}. For x e X and .v = /(x). choose affme open
neighborhoods U = Spec(^) and V = Spec(C) of x and s. respectively, so that
U C f~l(V). Then U x i■ U = Spec(/4 &>c /4) is an open neighborhood of a point
(x. x) in X xs X. Then Ax/s\u coincides with ALyr: U —> [/ x r U and is defined by
the ring multiplication m : A <g>c -A —> ^. w(« g> «') = ««'. Atyr induces, therefore,
an isomorphism between U and the closed subscheme of U x r U defined by the ideal
Ker(w) of A ®( A. Namely. ALy, is a closed immersion. Hence, we know that there
exist a family of aftine open sets { W; };l6A of A" x.v X and quasicoherent Ideals SA [X £
A) such that Ax/s(X) C U;.eA W}, and (A^-)-'^) - (Ar/S(«.^,/^) for
each X £ A. In particular. AX/s{X) is an open set of the closure Ax/s{X). (As shown
in Problem II.4.2. AX/s{X) does not necessarily coincide with AX/s{X).) Ax/s(X)
is a subscheme of X xs X in the following sense.
Definition 4.8. (1) Let X and Y be schemes. Y is called a subscheme of X if
there exist an open set C/ and a quasicoherent <f/ -Ideal S such that K is the closed
subscheme of U defined by J'. (If U = X. then K is a closed subscheme of X.)
(2) Let /: I -> S be a morphism of schemes. We say that / is a separated
morphism and X is separated over S by f if A^ is a closed immersion.
We can list the following properties of separated morphisms.
Lemma 4.9. (1) Let f': X —> 51 /?t' « morphism of schemes, let {VA}/€\ be an
open covering of S. and let I/;. = f~[{V/). Then f is separated if and only if so is
f\t;- V;.^V-Jbr each Xe A.
(2) A morphism f : X —> S with affine schemes X and S is separated.
(3) If f : X —> Y and g: Y —> Z are separated, so is the composite g • f : X —> Z.
(4) //" / : A" —> S iv « separated morphism and j : F —> A- w « closed immersion
(or an open immersion), then f • j: Y —> S « separated.
(5) //./: A" —> S iy « separated morphism. then any base change fs,: Xs> -* S'
by an S-scheme S' is a separated morphism.
(6) For morphisms of schemes f: X —* Y and g: Y —> Z. if the composite g • f
is separated, then so is f.
Proof. (1) Let /(2): X xs X —> S be the structure morphism. Then U}_ x r t/; =
(./(2')_'( I7;.)- Furthermore. {C/; xi; L^};.^ is an open covering of A" x.v X. and

4. SCHEMES AND ALGEBRAIC VARIETIES
89
AX/s(X) n (U>. xi; £/;.) = At//r (f/;) for each ). e A. Now the assertion follows
from the definition.
(2) Repeat the argument as developed just before Definition 4.8.
(3) By hypothesis. AX/Y: X -+ X x,Y X and AY/z' Y -+ Y Xz Y are closed
immersions. On the other hand. Ax/Z'. X -+ X xz X is the composite of the
morphism (AY/Z) xyX/r (/• f)z'■ X x Y X -+ X x.z X obtained as the base change
of AYjZ by (f.f)z'- X x?■ X -+ Y x7 Y and AX/Y. Since the base change of a
closed immersion and the composite of closed immersions are also closed immersions
(Problem II.4.3). Ax/z is a closed immersion. Thus, g ■ f is a separated morphism.
(4) Suppose j is a closed immersion. For an affine open set V — Spec(5) of X.
U := j~l(V) is an affine open set of Y provided U ^ 0. We can write U = Spec(^)
with ,4 = B/I. where/ is an ideal of B. Then the projection p\: U x r U = Spec(A®B
A) -+ U is an isomorphism. Hence, we know that the projection p\: Y xx Y -+ Y
is an isomorphism. Next suppose j is an open immersion. In this case, we can
argue with V C j(Y) to show that p\: Y xx Y -+ Y is an isomorphism. Since
P\ ' &r/x = 1 )'• AY/X is an isomorphism, too. Namely, we know that j is a separated
morphism. Then f ■ j is a separated morphism by (3) above.
(5) Since Ax>/s> = (Ax/s) xs S'. AXsl/s, is a closed immersion.
(6) / is the composite of Yf■: X -+ X xz Y and pi'. X xz Y -+ Y. where
Yf — (lx-f)z and pi is the natural projection. Now the diagonal morphism for T/
is an isomorphism: hence. Y f is a separated morphism. If we regard Y = Z xz Y.
then pi = (g • f) xz Y. so the projection pi is a base change of g • f. By (5).
Pi is therefore a separated morphism. Now we know by (3) that f is a separated
morphism. □
The significance of a morphism being separated will be explained after we define
an algebraic variety.
Definition 4.10. Let f: X -+ Y be a morphism of schemes.
(1) / is a quasicompact morphism if f~](V) is a quasicompact open set of X
for every quasicompact open set V of Y.
(2) f is a locally finitely generated morphism if for each point x of X there exist
affine open neighborhoods U = Spec(^) of x and V = Spec(5) of /(a)
such that U C f~l(V) and A is a finitely generated B-algebra.
(3) f is a finitely generated morphism if / is quasicompact and locally finitely
generated. We then call X a finitely generated Y-scheme. In particular, if
y = Spec(k) with a field k. we call a finitely generated F-scheme X an
algebraic scheme defined over k.
(4) Let X be an algebraic scheme defined over k. We call X an algebraic v.ariety
defined over k if X satisfies the following two conditions:
(i) X is irreducible and reduced, and X xSpec(/v.) Spec(£) (also denoted by
X (¾ k) is also irreducible and reduced, where k is an algebraic closure
of A:.
(ii) The structure morphism f: X -+ Spec (A:) is a separated morphism.
(5) Let Y be a subscheme of an algebraic variety X defined over k. We call Y
an algebraic subvariety of X if Y itself is an algebraic variety defined over
k.
Lemma 4.11. Let X be an irreducible reduced algebraic scheme defined over afield
k. Then the following assertions hold:

90
11. SC'HhMhS AND ALGhBRAlC VAR1ET1KS
(1) A' is a noetherian space.
(2) Let U = Spec(^) be an affine open set (^ 0) of X. Then the quotient field
Q(A) is a finitely generated extension ofk which is determined only by X independently
of the choice of U. {We call this field the function field of X over k and denote it by
k{X).) Then we have tr. deg,. k{X) = dim X.
(3) Let k he an algebraic closure of k. X ®k k is irreducible and reduced if and
only ifk(X) % k is a field, i.e.. k(X) is a regular extension of k. Then k(X X>/; k) =
k{X)®kk.
(4) Let x be a point of X. Then x is a closed point of X if and only if the residue
field k{x) of @x.\ if a finite algebraic extension of k. (We call x a /c-rational point if
k(x) = k.) When k is an algebraically closed field, the set X(k) of k-rationed points of
X possesses a local-ringed space structure which is induced naturally by X. Conversely,
the local-ringed space structure on X(k) restores the local-ringed space structure on X.
Proof. (1) There exists a finite affine open covering 11 = {U,}Ae,\ such that
AA := Y(U;.@x) is finitely generated over k. Since I/;, is then a noetherian space
and since 11 is a finite covering. X is a noetherian space.
(2) Let U = Spec(^) and V = Spec(5) be nonempty affine open sets. Then
U n V 7^ 0. A point .v of U n V corresponds to prime ideals p. q of A. B. respectively,
and &x.x = Av = 5q. So Q{A) = Q[&x^) = Q{B). Since we may take U so that
A is finitely generated over k. Q{A) is a finitely generated extension of k. With
the notations in (1). dimZ = max;6A dim U,. Since dim UA = diml^t/,. &L-J =
tr. deg,, k(X). we have dim X = tr. deg,, k(X).
(3) By the construction of X % k. U (g>k k = Spec(^ % k) is an open set of
X % k. Suppose X ®k k is irreducible and reduced. Then Q(A) C Q(A) (¾ k C
Q(A &k k). Since Q(A) (¾ k is integral over Q(A). Q(A) (g'k k is a field. Conversely,
suppose k(X) 0¾. k is a field. We shall show that X (¾ k is irreducible and reduced.
First of all. take U = Spec(^) as above. Then A (¾ k is an integral domain because
A 0/v k is a subring of k(X) ¢5/- k: in fact, k is a flat k-module. Next X (g>k k is
connected. If X <S)k k is not connected, there exists an affine open set U = Spec(^)
such that the inverse image /7J-1 (U) is not connected, where p\: X ®k k —> X is the
natural projection. Namely, there is an idempotent decomposition 1 = e\ + e2 in
A (g>k k such that ej = e\.e\ = e2. and e\e2 ~ 0. (Verify this claim.) This is a
contradiction because A ®k k is an integral domain. Once we know that X (¾ k is
connected, in order to prove that X (¾ k is irreducible and reduced, we have only
to show that the local ring &-_ for each point : of I &k k is an integral domain.
Put x = p\{:). and let U = Spec(/1) be an affine open neighborhood of a-. Then z
corresponds to a prime ideal q of A ($k k. and @z = (A <% k)q. Since A c$k k is an
integral domain as remarked above, so is &-_.
(4) Let a- be a point of X and choose an affine open neighborhood U = Spec{A)
of a- so that A is a finitely generated /:-algebra. Then a is a closed point of X if and
only if a is a closed point of U. Moreover, v is a closed point of U if and only if
k(x) is a finite algebraic extension of k ((1.1.15) and (1.1.10)). Suppose k = k. Then
U(k) endowed with the induced topology from U and the sheaf of local rings @L(k)
(which we denote by &i-1 (_(/>)) is an affine algebraic variety, and (U(k).@L!(k)) restores
(U.@i) as a local ringed space (1.1.24). Since X has a finite affine open covering as
specified in the above proof of (1). the induced topology on X(k) from X restores
the topology on X. Define @X(k) % @x(k)\u,{k) = @L,(k)- Trien {X{k)-@x(k)) *s a
local ringed space and restores (X.&\). O

4. SCHEMES AND ALCihBRAlC VARlhTlES
91
In view of (4) of Lemma 4.11. we may confuse deliberately an algebraic variety
[X.&x) with (X(k).0y{ki) when k is an algebraically closed field. As the sets of
points. X differs from X{k) in the point that X contains the generic points of all
closed algebraic subvarieties of dimension > 1. Now let X be an algebraic scheme
defined over a field k with the structure morphism /: X —> Spec(&). We are going
to explain the significance of f being a separated morphism in terms of discrete
valuation rings. For this, we need some preparation.
Lemma 4.12. Let A be a finitely generated k-algebra domain, let K be its quotient
field, and let L be a finite algebraic extension. Then for an arbitrary prime ideal p of
A. there exists a discrete valuation ring & of L such that & dominates Ap. Namely.
& D Ap and m n Ap = p^p. where m is the maximal ideal of &. We express it by
&>AP.
Proof. Let (x\ x„) be a set of generators of p. i.e.. p = (.vi x„). and
let B = A[x2/x\ v„/vi]. Then pB = \\B. Let R be the integral closure of
B in L. By (1.1.34), R is a finitely generated /:-algebra domain. Let T = A - p.
Then RT is a normal ring, and pRr = x\RT. Let ^3 be a prime divisor of .vi/?•/■.
Then ^3 has height 1 and (Rr)<p is a discrete valuation ring of L (1.1.33). Since
<P(/?7')tp HAp is a proper ideal of Ap containing pAp. we have ^3(^/ )«p C\AP= pAp.
Hence. (Rr)<$ > Ap. Thus, we have only to take & = (RT)<p. O
Let (if.m) be a discrete valuation ring. The prime ideals of & are (0) and the
maximal ideal m only. Namely. Spec(^f) consists of two points, the ideal (0) being
the generic point and m being the unique closed point. Let g: Spec(^f) —► X be
a morphism of schemes, and let g((0)) = .v and g(m) = x'. Then there is a local
homomorphism tp: @x.\' —► & associated with g. (g is factored by an affine open
neighborhood U = Spec(^) of .v'.) A prime ideal q := <£>~'(0) corresponds to
the point x. and @x.\'l<\ = &y.\-<- where Y := ({^})red. Moreover. &Y.Ki < &.
Furthermore, tp induces a homomorphism from the residue field k(x) = &x.x/vnx.-,
to the quotient field Q{@) of &. Conversely, the following result holds.
Lemma 4.13. Let X be an algebraic scheme over a field k. let x e X. let Y =
({A"})red- cmd let x' S Y. Let L be a finite algebraic extension of the function field
k{Y). Then there exists a discrete valuation ring & of L such that & > @y.si.
Proof. Since Y is an algebraic scheme over k. there exists an affine open
neighborhood Spec(^) of x' in Y such that A is a finitely generated /:-algebra domain.
Let p be a prime ideal of A corresponding to x'. Then, applying Lemma 4.12. we
find a discrete valuation ring & of L such that & > Ap = @y.x>- D
Under the circumstances described just before Lemma 4.13. we say that x' is
a specialization of x along & (or dominated by &). and write it as x —► x'. The
choice of a discrete valuation ring & which dominates a specialization x —► x' is not
unique. Even worse, if we give a point .v of X and a discrete valuation ring & with
k(x) C Q{&). a specialization x —> x' along & is not necessarily unique (cf Theorem
4.15 and Problem II.4.2).
Lemma 4.14. Let X be an algebraic scheme over a field k. and let Y be a subscheme
of X. Then the following two conditions are equivalent:
(1) Y is a closed subscheme.

92
11. SCHEMES AND ALGEBRAIC VARIETIES
(2) If x S Y and x —► .v' is a specialization along a discrete valuation ring &. then
x' e Y.
Proof. (1) implies (2). There exists a local homomorphism tp: &x.x, —► & such
that a- corresponds to a prime ideal ip~l(0). Hence, x' e {x} C F.
(2) implies (1). Suppose x' e F. Then there exists a point x of Y such that
a' e {a}. By Lemma 4.13. there exists a discrete valuation ring^f of k(x) and a local
homomorphism p: @x x> -^ &■ Hence, a —► a'. By (2). a' e F. Hence. Y = Y. D
With this preparation, we shall prove the following.
Theorem 4.15. Let X be an algebraic scheme over k. and let f: X —» Spec(fc)
be the structure morphism. Then the following conditions are equivalent:
(1) f is a separated morphism.
(2) Given a discrete valuation ring & and specializations x —> x' and x —> x" along
& such that the induced homomorphisms of the fields k(x) —► Q(&) are identical, then
we have x' = x".
Proof. (1) implies (2). As explained before Definition 4.8. the image of X by the
diagonal morphism hX/k is a closed subscheme of X xk X. Note here that X xk X is
an algebraic scheme over k. Suppose two specializations a —► x' and a —► x" along ^
are respectively given by morphisms of schemes g. h: Spec(^f) —► X. Then we have
the morphism (g.h): Spec(^f) —► X xk X. Let _y be the image of the closed point
m of & by (g.h). Then p\(y) = x' and p2(y) = x". where p\ and /?2 are the natural
projections of X x/. X to X. Let .K = Q(&). and let ^: Spec(.K) --> Specif) give the
generic point of Spec(^). Then (g.h) ■ rj = AX/k • ix. where /v: Spec(.K) —► X is the
morphism defined by the homomorphism $V..Y —> /c(a) —> Ar. We have y e {(.v. a)}.
and >• e hX/k(X) by the hypothesis in (1). So x' = p\(y) = pi(y) = a".
(2) implies (1). Suppose _y s {(x.a)}. By Lemma 4.13. there exists a discrete
valuation ring & such that (a. a) —► _y is a specialization along ^. Namely, there
exists a morphism /: Specif) —► X xk X such that /((0)) = (a.a) and /(m) = y.
Let g = p\ • I. h = pi'l. j?(tn) = a', and A(m) = a". Then a —> x' and a —► a" are
specializations along &. By (2). we have a' = x". so g — h. Namely. / is factored by
&x/k- Then _y e AX/k(X). So A .*-//,- is a closed immersion, and X is thus separated
over /:. □
We have given some explanations on what algebraic schemes and algebraic
varieties are. We shall define some of their more specific properties.
Definition 4.16. (1) Let k be a field. A separated algebraic scheme X over k
is called proper over k if it satisfies the following condition:
Let & be a discrete valuation ring containing k. and let K be its quotient field.
Given a point a of X and a field homomorphism from the residue field k(x) to K.
there exists a specialization a —► x' along &.
If we denote the morphism induced by & °-^ K by r\: Spec(.K) —> Spec(^f). the
above condition is equivalent to saying that a mapping
Honu(Spec(<?). X) -► Hom,,(Spec(/0- X). g i-> g ■ rj
is surjective for every discrete valuation ring &.
(2) When X is an algebraic variety defined over k. we call X a complete algebraic
variety if X is proper over k.

4. SCHEMES AND ALGEBRAIC VARIETIES
W
(3) Let f: X —* Y be a morphism of schemes. We say that / is a closed morphism
if f(T) is a closed set of Y for every closed set T of X. We say that f is a universally
closed morphism if the base change f7 '■ X7 —► Z of / is a closed morphism for every
F-scheme g: Z —► Y.
(4) We call /: X —► Fa proper morphism if / is finitely generated, separated
and universally closed.
We have then the next result.
Theorem 4.17. Let X and Y be separated algebraic schemes over a field k. and
let f': X —> Y be a morphism of schemes. If X is proper over k. then f is a universally
closed morphism. Hence, f is a proper morphism. In particular, the structure morphism
X —► Spec(/c) is also a proper morphism.
Proof. (I) We shall prove that fz '■ Az —> Z is a closed morphism only in the case
where Z is a F-scheme and an algebraic scheme over k. J/ := I xyZ isa closed
subscheme of X xk Z. In fact, since F is separated over k the diagonal morphism
AY/k- F —+ F xa- F is a closed immersion. The natural morphism X xYZ ^ X x/, Z
is obtained from AY/k as the base change by X x* Z —» F xk F. So with the structure
morphism/?: X —> Spec(&). if we show that/?z : X xk Z —► Z is a closed morphism.
then the restriction fz of /?z onto Ix^Zisa closed morphism. Hence, we may
assume that F = Spec(/c) in the statement.
(II) Let /: X —► Spec(/c) be anew the structure morphism of X. We shall show
that the base change g : = fz '■ X xk Z —► Z is a closed morphism for an algebraic
scheme Z over /c. Let T be an irreducible closed subset of X xk. Z. and consider the
reduced, closed subscheme structure on T. Let S = g(T). We regard S as a reduced,
closed subscheme of Z. Let t be the generic point of T. and let s = g{t). Then 5
is an irreducible scheme with the generic point s. Let s' be an arbitrary point of S.
Then there exists a discrete valuation ring & such that Q{@) = k(t) and & > @s.s'-
In fact. k(t) is a finitely generated extension of k(s). Let A = @s.s'[£,\ £/]• where
{<jfi £,} is a transcendence basis of k(t) over k(s). Then &(?) is a finite algebraic
extension of the quotient field K of A. Let p be a prime divisor of m.y..V'A Applying
Lemma 4.12. we find a discrete valuation ring & of k(t) such that & > Ap. Then
^1 > @s.s'- Let /?i: X xk Z —> Z be the natural projection, and let .v = p\{t).
Then k(x) ^-+ k(t) = 2(^)- Hence, by the hypothesis that X is proper over k.
there exists a specialization x ■—» x' along ff. So there exists morphisms of schemes
h: Spec(^) -^ X and /: Spec(^) -> S such that h{(0)) = x. h(m) = x'. /((0)) = .v.
and /(m) = .v'. where m is the maximal ideal of & and is identified with the closed
point of Spec(^f). Hence, there exists a morphism q := (h.l): Spec(^) —» X xk Z
such that ¢/((0)) = f. Let f' = q{m). Then r' G {t}. Since T is a closed subset.
t' e T. So s' = g(t') e g{T). We therefore have 5 = g(T). Namely. g(T) is a
closed subset of Z.
(III) Once we know that /: X —► F is a universally closed morphism. we know
that / is a proper morphism because f is finitely generated and separated (cf. Lemma
4.9 (6)). ' □
A morphism f: X —> F between irreducible and reduced schemes is called
dominant if f (X) = Y.
Theorem 4.18. Let X and Y be separated algebraic schemes over afield k.
Suppose X is irreducible and reduced. Let f : X —* Y be a proper morphism. Then f\&x

94
II. SCHEMES AND ALGEBRAIC" VARIETIES
is a coherent &y -Module. If X is a complete algebraic variety defined over k. we have
Y{X.@x) = k. Furthermore, for a coherent &\-Module 9~. R'f\&~ is a coherent
&Y-Module for each i > 0.
Proof. (I) Let Z be a closed set f{X) in Y endowed with the reduced closed
subscheme structure. Then / decomposes to a product of g: X -+ Z and a closed
immersion r. Z ^-+ Y. where g is a dominant, proper morphism (cf. Lemma 4.19
below). If g*&x is a coherent ^?z-Module. then it is easy to show that f*@x is
a coherent @Y-Module. Hence, replacing Y by Z. we may assume that Y is an
irreducible, reduced scheme and f is a dominant morphism.
(II) Let k(X) and k(Y) be the function fields of X and Y over k. respectively.
If £, is the generic point of X. then n = f{£,) is the generic point of Y because
f is dominant. Moreover, since k(X) = k{£,) and k(Y) = k{n). f induces the
field homomorphism k{Y) ^-+ k(X) by which we regard k(Y) as a subfield of k(X).
Evidently. k(X) is a finitely generated extension of k(Y). and the algebraic closure
L of k(Y) in k(X) is a finite algebraic extension (1.1.5). Meanwhile, the property
that f*@x is a coherent cfy-Module is a local property on F. we may assume that
Y is an affine scheme, by replacing Y and X by an affine open set Spec(^) and
/-1(Spec(/4)). respectively (cf. Lemma 4.19 below). Furthermore, we may assume
that A is a finitely generated /c-algebra domain. Note that Q{A) = k(Y). The integral
closure B of A in L is a finitely generated ^-module (1.1.34). On the other hand.
f*@x is a quasicoherent ffy-Module (problem II.4.9). Hence. f^&x = {Y{X .&x)Y
and Y{X.@x) = five* &x.x- We shall prove in the next step (III) the following two
assertions:
(1) Let {^;};eA exhaust all discrete valuation rings such that Q{&>) = k(X) and
A ^0,, Then 5 = C\x^0k.
(2) For each &-,., there exists a point x of X with &x > @x.\.
Suppose these two assertions are proved. Then f]- &-,_ D C\xeX @x..\ (cf. Lemma
4.12). Namely, B D Y(X.@X)- Since B is a finitely generated .4-module and A is a
noetherian ring, Y(X.&x) is a finitely generated ^-module.
(III) We shall prove the above assertions (1) and (2). Since &-A is integrally closed,
we see that B c f].^. Take £, e C\f9x. Suppose £, i B. Then d;-1^-1] is a
proper ideal of ^[£-1]. Here we repeat the argument in the step (II) of the proof of
Theorem 4.17 to show that there exists a discrete valuation ring (&. m) of k{X) such
that & D A[i-[] and m D £_~XA[^~X]. Then & belongs to {^.}A6A. but ¢¢0. This
is a contradiction. Hence, B = f)x^- Next we shall prove the assertion (2). Let
@ =@x-K = k(X). and S = Spec(^). Since ffOA.we have a morphism h : S -+ Y.
Let g: Z := Xs -+ S be the base change of f: X -+ Y by h. Let r\: Spec(.K) -+ S
be the morphism corresponding to the inclusion & ^-+ K. Since K = k(X). there
exists a morphism a: Spec(A^) —> Z such that g • a = n and p\ • o is a dominant
morphism. where p\: Xs -+ X is the natural projection. Since f is a universally
closed morphism. g is a closed morphism. Let £, be the image of Spec(A^) (which
consists of one point) by a. and let T = {£}. Then g(T) = S because g(T) is a
closed subset containing the generic point of S. So there exists a point : of T such
that g{z) is the closed point of S. Consider the reduced closed subscheme structure
on T induced by Z. Then @T- > & and @T. is a local domain1 whose quotient field
is K. Hence. &T- = &. In fact, if a £ @j- — &'. then a~] e m(=the maximal ideal
' A local ring which is an integral domain.

4. SCHEMES AND ALC3EBRAK" VARIETIES
95
of^f). so «_1 S m-r.: and 1 = « • a~x s my.:. This is a contradiction. Let x = p\{z).
Since/?i|v : T —* X is a dominant morphism. we have &T-_ > &x.\- So. & > &x.\-
Thus, we proved the assertions (1) and (2).
(IV) Let X be a complete algebraic variety defined over k. and let/?: X —> Spec(^r)
be the structure morphism. Then />*<^V is identified with Y{X.&\). We know that
Y{X.&x) is a finitely generated k-module contained in k{X). Hence. Y{X.&X) is a
finite algebraic extension of k. Then we conclude that Y{X.&X) = k because k is
algebraically closed in k{X).
We do not give a proof of the last assertion of the theorem. However, the reader
will encounter a related result in Theorem 5.13 later. For a rigorous proof, we refer
to [3, EGA. Chapter 3], □
In the above proof, we used the following result.
Lemma 4.19. (I) A closed immersion is a proper morphism.
(2) If X -+ Y is a proper morphism then the base change f: X' := XY, —> Y'
by a morphism of schemes g : Y' —> Y is a proper morphism.
(3) IfX —> Y and g : Y —+ Z are proper morphisms. then the composite g • f : X —►
Z is a proper morphism.
(4) Let S be a scheme, and let /',■: X/ —► Y, (i = 1. 2) be a proper morphism of
S-schemes. Then f\ xs fi'. X\ x.y %2 —► Y\ x,y Yi is a proper morphism.
(5) Suppose the composite g • f of two morphisms of schemes f: X —► Y and
g: Y —> Z k a proper morphism. Then f is a proper morphism provided g is a separated
morphism.
Proof. Suppose (1). (2). and (3) were proved. Then (4) and (5) follow from
them. In fact, the morphism f\ xs f2 in (4) is the composite of the two morphisms
which are obtained from f\ and fi as the base changes. Hence. f\ x <,- fj is a proper
morphism. In the case of the morphism f in (5). we have f = p2- Y t as in the proof
of Lemma 4.9 (6). where p2 and Y f are the base changes of proper morphisms. So
f is a proper morphism. We shall prove (1). (2). and (3). In order to show that a
morphism f: X —* Y is a proper morphism. we have to verify all three properties that
f is quasicompact. locally finitely generated and universally closed. So we shall show
that the morphisms considered in (1). (2). and (3) satisfy each of these properties.
However, since the requirement that a morphism be locally finitely generated is easy
to check, we omit it.
(1) Let f: X —► Y be a closed immersion. Since a closed subset of a
quasicompact set is also quasicompact. f is a quasicompact morphism. Any base change of
a closed immersion is a closed immersion (Problem II.4.3). and a closed immersion
is clearly a closed morphism. Hence. /' is a universally closed morphism.
(2) Let U' be a quasicompact. open set of Y'. For a point y' £ U' and
g(y') S Y. we may take affine open neighborhoods U of g(y') and W of y' so
that W c U' ng~'(£/). U' is covered by finitely many If's of this kind. So we
have only to show that f'~x{W) is a quasicompact. open set. Hence, it suffices
to show that X' is quasicompact if Y' and Y are affine schemes. By the proof of
Theorem 4.5 concerning the existence of fiber product. { V,x Y Y'}AeA is a finite affine
open covering of X' if {F;.};i6a is a finite affine open covering of X. which exists
because X is quasicompact. Since an affine scheme is quasicompact. a finite union
of affine schemes X' = \J^(V;. x Y Y') is also quasicompact. Thus. /*': X' —► Y' is a
quasicompact morphism. On the other hand, it follows readily from the definition

96
11. SCHEMES AND ALGEBRAIC VARIETIES
that /' is a universally closed morphism.
(3) It is easy to verify that g ■ f is a quasicompact morphism as well as a
universally closed morphism. □
Definition 4.20. Let f : X —► Y be a morphism of schemes, f is said to be an
affine morphism if there exists an affine open covering { K;.};eA such that f~y{V-,) is
an affine open set of X for each X e A. If we write V-,. = Spec(^;) and f~x{V?) =
Spec(5;). B, is an ^-algebra. So we may view B,. naturally as an ^.-module. If
we choose the covering {K;.};.6A so that B,_ is a finitely generated ^.-module, we say
that f is & finite morphism.
We now have the following.
Theorem 4.21. Let X and Y be separated algebraic schemes over a field k. For
a morphism f: X —> Y, the following two conditions are equivalent to each other:
(i) /' is a finite morphism.
(ii) f is a proper and affine morphism.
Proof, (ii) implies (i). There is an affine open covering {V;}a(ea of Y such
that f~l(V;_) is an affine open set of X for each X e A. The base change /;_ =
/ xy V-,.\ f~y{V,) —► V-,_ is a proper morphism (Lemma4.19). Hence, we have only
to show by the condition (ii) that A is a finitely generated 5-module if X = Spec(^)
and Y = Spec(5). But this is immediate from Theorem 4.18 because f%<9\ = A is
a coherent 5-Module.
(i) implies (ii). There exists an affine open covering {V;.}x&a such that f~l(V;.)
is an affine open set of X for each IgA and A-,, is a finitely generated 5;.-module,
where V,. = Spec(5;.) and f~\V,) - Spec(^;.). If /;. - / x Y V,_: /-'(K;.) -> Vk
is a proper morphism for each X e A. then f is a proper morphism. Since /;_ is
quasicompact and separated (Lemma 4.9), it suffices to show that /;_ is universally
closed. Namely, we have only to show that if X = Spec(^). Y = Spec(5). and A is a
finitely generated 5-module. then f is a universally closed morphism. Let g: Z —► Y
be any morphism of schemes, and let {Z,},e/ be an affine open covering of Z. Then
the base change f,■ = f x Y Z,>: X x Y Z, —+ Z, is a finite morphism. In fact, if
Z, = Spec(C,), then X xY Z, = Spec(^ ®b Q) and A <S>b Q is a finitely generated
C,-module. Moreover, if/,- is a closed morphism, then fz~fxYZ: X xY Z —► Z
is a closed morphism. So all in all. it suffices to show that if X and Y are affine
schemes as above and A is a finitely generated 5-module. then / is a closed morphism.
Let T be a closed subset of X. T is an affine scheme, and if T = Spec(C). then C is
a finitely generated 5-module. Hence, replacing X by T we have only to show that
f{X) is a closed set of Y. Since f(X) = {q € Spec(5): for some p e f(X). p C q}.
it suffices to show that if q e Spec(5) satisfies the condition p C q with p = V^'^P
for some <J3 e Spec(^). then q = </?_1(£2) with £2 € Spec(^). where ip\ B —> A is a
ring homomorphism with / = "tp. Replace A and B by ,4/^J and 5/p. respectively.
Then the induced ring homomorphism <p\ B/p —» ^/¾} is injective, and A/?(S is a
finitely generated 5/p-module. The prime ideal Q which we require corresponds to
£2 S Spec(^/*p) such that <^_1 (£2) = q/p. Hence, we may assume from the beginning
that A and B are integral domains and B is a subring of A. Then the prime ideal
£2 exists by (1.1.18). □
If /: X —> F is a finite and dominant morphism of algebraic varieties defined
over a field k. we say that A" is & finite covering space (or a finite covering, for short) of

4. SCHEMES AND ALGEBRAIC VARIETIES
97
Y. As an example of finite morphism, we shall explain the normalization morphism
of an algebraic variety.
Definition 4.22. We call a scheme X a locally noetherian scheme if each point
x of X has an affine open neighborhood Spec(^v) such that Ax is a noetherian
ring. Furthermore, if X is quasicompact. we call X a noetherian scheme. A locally
noetherian scheme X is called a normal scheme if the local ring &x.x of each point
x is a normal ring, i.e., a noetherian integral domain which is integrally closed. An
algebraic variety X defined over a field k. which is a normal scheme, is called a normal
algebraic variety.
We have the following result.
Theorem 4.23. Let X be an algebraic variety defined over afield k. and let K be
its function field. For a finite algebraic extension L of K. there exist a normal algebraic
scheme Z defined over the field k and a dominant finite morphism f: Z —► X satisfying
the following conditions:
(\)L = k(Z).
(2) For a normal algebraic scheme Z' over k with L = k(Z') and a dominant finite
morphism f: Z' —► X. there exists an isomorphism g: Z —» Z' such that f = f • g.
Proof. (1) We can take an affine open covering {U,}xet\ such that AA: = T{Ux- &x)
is a domain which is finitely generated over k2 Note that since X is separated over k.
U;.M :— Ux H UM is also an affine open set (cf. (11.3.12)). Note also that K = Q{A}).
Let RA be the integral closure of A> inL. By (1.1.34). R, is then a finitely generated k-
domain and a finitely generated ^-module. Let fx '■ W-K —+ Ux be the finite morphism
associated with the inclusion A-,_ ^-> Rk. where W-K = Spec(/?;.). If fi 7^ L then the
coordinate ring AAfl := TiU^.^x) is given as AxM = fl-GL<„, @x.x-
Let RxM be the integral closure of AxM in L. We claim that Spec(/?^) is naturally
identified with /r'([/^). Since /~'(^) = Wk xu, U^ = Spec(/?;. ®A/ A;4l).
fJX{Ui.ii) is an affine open set of W;_. Hence, the coordinate ring R; ®A/ AxM is
an integrally closed, integral domain C\wsf~\u \@wm<- and Rx ®a, A-,^ is a finitely
generated ^^-module because R, is a finitely generated ^;-module. Thus, as a
subring of L. RxM is identified with R; ®a, A^. Under this identification, we have
/r'(^) = Spec(^).
Similarly, /"'(C/;.^) is identified with Spec(/?^). Hence, there exists an
isomorphisms^: /r'(C/^) -^f~l{U^). Furthermore, £/^,, = UxDUMr]Uv {L/i. v <E A) is
an affine open set, and if AxMV and Rxf,r denote respectively the coordinate ring of £/^,,
and the integral closure of A^, in L, then fyl(U^v), /^(¾..). and /,71 (£/^.0
are identified with Spec(/?^,v). Since these identifications are given by the natural
identifications of R; ®Ai A-/41V, RM CS>Afl AxMv and Rv <g>Ar A-,^, with the subring RxMV
in L. we have a relation ,v„;. = sVfl ■ sMx- Since sxx = id (A S A), there exists by Lemma
4.3 a scheme Z and a morphism f: Z —► X such that for each 1 e A, Wx and
fx are identified with an open set /-1 (£/;.) of Z and f\w,- respectively. Then by
the construction. Z is a normal algebraic scheme over k and f is a dominant finite
morphism.
(2) Let {£/;.};.gA be an affine open covering of X as above. Then f'~x{U\)
is an affine open set of Z' by Lemma 4.24. Furthermore, the coordinate ring R'}
-We call it a finitely generated i-domain.

98
11. SCHEMES AND ALGEBRAIC VARIETIES
of/' '(£/;.) is given as R1- = f1-e/-'-'(r )®7-' - and is an integrally closed, integral
domain which is a finitely generated ^-module. If we view R'; as a subring of L.
then we have R; = R'- because both R; and R'; are the integral closure of A-, in L.
So there exists an isomorphism g-,.\ /"'(£/;.) ^ //_'(£/;.) such that / = /' • g;. on
/~'(£/;.). Let /?; be the coordinate ring of f'~l(U;.fl). Then /?; . as a subring of
L. coincides with the integral closure R/4, of A-,^. This implies that the restrictions
of g,. and g^ onto /_1(£/;^) are equal. By virtue of Lemma 4.3 (2). we conclude
the existence of an isomorphism g: Z —» Z' such that g, = g|/--i(t') f°r eacn ^ e ^
and/=/'-g. ' □
In the above proof, we made use of the result below.
Lemma 4.24. Lt'f / : Z —► X />c cot «#'w morphism. Then /"'(£/) is an affine
open set for every affine open set U ofX. Iff is a finite morphism. then Y(f~{{U).$x)
is a finitely generated T{U. @x)-rnodule.
Proof. We readily see that /|/--i(t> /"'(£/) —> £/ is an affine morphism. In
fact, let {£/;.};.eA be an affine open covering of X such that {/~'(£/;.)};.eA is an
affine open covering of Z. Then for an affine open set V with V C U-,C\U (for
some X s A). /"'(^) is an affine open set of Z and £/ is covered by such affine
open sets V\ If / is a finite morphism and r(/_1(£/;.).@z) is a finitely generated
r(£/;.^)-module for each X e A. we can argue as above that / |/ -i(l) is also a finite
morphism. Hence, it suffices to prove the lemma under the hypothesis U = X.
Let A" = Spec(^). 8 = r(Z.^z).and Y = Spec(B). Then the morphism / is the
composite of the two morphisms h ; Z ~> Y and g: Y —► X (Lemma 4.4). Meanwhile,
if {£/;.};.£a is an affine open covering of X as above. W-,. = /~'(£/;.). and /;. = /|ie>
then /;^z|t- = (/;),<%, (Lemma 4.7). If we put /L = Y{U-,.@X) and 5; =
Y(Wk.&z). then (//.),^, = 5;. and (.f;.)*^^ is thus a quasicoherent $/-Module.
So /*(^z is a quasicoherent ^-Module (cf Problem II.4.9). Therefore, f *@z = B
(Theorem 3.12). Since g~\U?) = Spec(B ®A A-,) and {B ®A A,)'- = f*&z\v,
(Lemma 3.22 (3)). and since W-,. is an affine scheme and f*&z\v, = {f>)*@w,- we
know that the morphism h induces an isomorphism between W). = /_1 (£/;.) and
g~[(U;.). So h : Z —* Y is an isomorphism, and thus. Z is an affine scheme.
We may assume that for the affine open covering {£/;.};.ga- the set A is a finite
set. £/; = D{t;) with t-,. e A. and B, is a finitely generated ^-module. Take a
finite system of generators {b-,-/. i e I?} of the ^;.-module B, with />,;. e 5. Then
{/>,/.:' 6 /• X e A} is a finite system of generators of the /1-module B. The proof is
similar to the proof of Theorem 3.12 (1). □
The scheme Z and the finite morphism /: Z —► X. whose existence was proved in
Theorem 4.23. are called the normalization of X in L and the normalization morphism.
respectively. If L = K. then Z is an algebraic variety defined over k. We then call
Z simply the normalization of A\
In algebraic geometry, rational mappings of schemes, which we discuss below,
are as significant as morphisms of schemes.
Definition 4.25. Let X and Y be S-schemes. Consider the set R consisting of
pairs (£/./). where U is a dense open subset of X and an S-morphism /: £/ —> Y.
Two elements (£/./) and (F. g) of R are equivalent to each other, denoted (£/./) ~
(V.g). if there exists a dense open subset W C £/ n V such that /|^ = g\w- This
is an equivalence relation on R. A pair (£/./) or its equivalence class is called an

4. SCHEMhS AND ALGEBRAIC VARIETIES
99
S-rational mapping from X to F and expressed by f: X ■ ■ ■ —> 7. We say that /
is defined on U. If there is no fear of confusion, we denote it also by /': X —> F.
For a point a- of X. we say that / is defined at x if there is a pair (V.g) equivalent
to ([/. /) such that x e V. The set of points x of X where / is defined is a dense
open subset of X. which we call the domain of definition of /: X ■ ■ ■ —> 7.
Lemma 4.26. Lcf X />e a reduced scheme, let Y be a scheme separated over S.
and let j': X ■ ■ ■ —> F />f an S-rational mapping. Let Uq be the domain of definition
of f. Then there exists an S-morphism /o: U0 —> Y such that (t/o./o) represents the
equivalence class of f.
Proof. If f\: U —> Y and /2: t/> —> Y are S-morphisms representing the class
of f. then there exists a dense open set V C. U\ n U2 such that f\\y = /air- Put
A" = t/i n U2 and /,' = /~,|X' (1 = 1.2). where /,': X' -> Y are S-morphisms.
We claim that f[ = f2. Put g = {f[.f'2)\ X' ^ Y xs Y and h = /,|,- = /2|>-.
Then g|j- = Ar/S • «. Since Y is separated over S. AY/s is a closed immersion and
the scheme {X'.g) xYx.sy (Y.AY/s) is. therefore, a closed subscheme of X' which
contains a dense open set V. Here we used the notations {X'.g) and {Y.AY/s) to
emphasize that X' and Y are schemes over Y x s Y by the respective morphisms g
and Ar/S. Since X' is reduced. (X'.g) x Y*s y {Y.AY/S) is. thus, identified with X'.
This means that the S-morphism h : V —> F is extended onto X' and ^ = AY/s • h.
So « = /| = /j as claimed above. By Lemma 4.3. f\ and /2 are then extended
to an S-morphism /3: f/i U U2 —> Y and (t/i U U2.fr) represents the class of f.
Repeating this argument, we derive the existence of an S-morphism (t/o/o) which
represents the class of /. □
Hereafter, we are mainly interested in the case S = Spec(fc), where k is a field.
Lemma 4.27. Let k be afield, let X be an irreducible reduced k-scheme. and let Y be
an algebraic k-scheme. Then there is a one-to-one correspondence between the following
two sets: {k-rational mapping f: X ■ ■ ■ —> Y} —+ {k-morphism fn: Spec k{X) —> F}.
The correspondence assigns to f the composite fn : Spec k {X) -A X ■ ■ ■ -^ Y. where n
is the generic point of X and in: Spec k{X) —> X is the natural morphism which maps
the unique point of Spec k{X) to n.
Proof. Suppose {U.f) represents the class of /: X ■ ■ ■ —> F. Then in
decomposes as SpecA^X) ^4 U ^ X and fn = / • in,u. If {U\.f\) ~ ([/2-/2)- we know
readily that f\ • in.ut = /2 • Vu,- Hence, the A:-morphism fn: SpecA-(A') —> F is
determined by /. Conversely, given a k-morphism g: Speck{X) —> F. let j be the
image of the unique point of Spec k{X) by g. Let V = Spec(5) be an affine open
neighborhood of y such that B is a finitely generated A:-algebra. Then g is given by a
/:-algebra homomorphism ip\ B —> A:(X). Write B = /:[^i />„] and let a, = tp{bj).
Then there exists an affine open set U = Spec {A) of X such that a, e ^4 for each i.
Thereby, g decomposes as
g: Speck{X)'^ uXvUY.
where h is the morphism associated with <p: B —> A and j is the natural open
immersion. Then g = fn. where we put / = /•«. □

100
II. SCHEMES AND ALGhBRAlC VARIETIES
Corollary 4.28. Let k be a field, and let X be an irreducible reduced k-scheme.
Then there is a one-to-one correspondence between the set of k-rational mappings
f : X ■ ■ ■ —> Aj. and the set of the elements of the field k{X).
Proof. /;/: Spec/^A") —+ A], = SpecAr[?] is determined by a A>algebra homo-
morphism ip: k[t] —> k(X) which is determined by the image <p(t) of t. D
Now let X and Y be algebraic varieties denned over a field k. and let /: X >
Y be a /:-rational mapping. Suppose f is represented by a pair of an open set U
of X and a morphism fL-. U -+ Y. Since Y is separated over k. the morphism
(l{/-/i'): £/ —> £/ x* T is a closed immersion. So its image TfL. is viewed as a
closed subset of U xk Y. Let T/ be the closure of TfL. in X xk Y. We call Tf the
#/•«/?/; of the rational mapping f. Since X and 7 are algebraic varieties. X xk Y is
an irreducible reduced scheme. (Let U = Spec(v4) and F = Spec(B) be affine open
sets of X and Y. respectively. Then U xk V is an affine open set of X xk Y and
A <% B is a subringof k(X)(g)k k(Y). Since k{X) and k(Y) are regular extensions of
k. k(X) (¾ Ar(r) is an integral domain. So A <g>k B is an integral domain. Compare
the proof of Lemma 4.11 (3).) If one considers on I~/ the reduced closed subscheme
structure of X xk Y. T f is an algebraic variety defined over k. Let p: T f -+ X
and a: T/■ —> 7 be the morphisms obtained by restricting onto Tf the projections
from X xk Ftol and Y. The sets T fv and U are respective open sets of T f and
X and isomorphic to each other by p. Hence, we can define a rational mapping
p~x: X ■ ■ ■ -+ Tf by U ^+ TfL ^ Tf. The composite of rational mappings q • p~]
is defined in the natural fashion, and we have q ■ p~{ = f.
If /V : U -+ Y is a dominant morphism, we call f a dominant rational mapping.
Then /1/(/7) is the generic point of Y if n is the generic point of X. So the field
homomorphism f\,: k(Y) -+ k(X) is determined. This homomorphism is
independent of the choice of a pair (£/. f u) which represents the class of/,. So we denote it
by /*: k(Y) -+ k(X). Conversely, let ip: k(Y) -+ k{X) be a field homomorphism.
Choose an affine open set V =■ Spec(B) of Y so that B is a finitely generated k-
algebra. Then Q(B) =k(Y). and there exists a Ar-rational mapping f y : X ■ > V by
Lemma 4.27. so that /*, = ip. We may consider the composite / = j ■ f \-: X ■ ■ ■ -+ Y
instead of //. where /: V °-» Y is the natural open immersion. Hence, there is a
one-to-one correspondence between the set {/: X ■ ■ ■ -+ Y; a dominant ^-rational
mapping} and the set {/*: k{Y) -+ k{X)\ a field homomorphism}. As a special
case, if / *: k{ Y) -+ k(X) is a field isomorphism over k. then there exists a dominant
^-rational mapping g: Y ■ ■ ■ -+ X such that g • f = \x and f • g = 1 r- If / is
represented by (U. fu). the composite g • f of rational mappings is defined if the
domain of definition of g has nonempty intersection with fu{U). We say that the
rational mapping / is a birational mapping if /*: k(Y) -+ k(X) is an isomorphism.
We also say that X and Y are birational to each other if there exists a birational
mapping /: X ■ ■ ■ -+ Y.
Example. Let X = X2k = Spec/:[x.j]. and let Y = A2k = Spec£[?. u\. Define a
rational mapping /: A1 • • • —> Y by the field isomorphism ip: k(t. u) -+ k(x.y) such
that ip(t) ~ x and <p(u) = y/x. Then the domain of definition of/ is A^ — {(0. b)\ b e
k}. The rational mapping /-1: Y ■ ■ ■ —► X is given by i//(x) = ? and y/(y) = tu.
and /-1 is a morphism.
In the case where /: X ■ ■ ■ -+ Y is a dominant rational mapping, we consider
k{Y) as a subfield of k(X) through /*. where we also consider k{X) = k{Tf). For

4. SCHEMES AND ALGEBRAIC VARIETIES
101
points x £ X and y 6 F, we say that x corresponds to y by f if there exists a point
z of T/ such that x = p(z) and y = q{z). We denote this relation temporarily by
x ~f y. Let F be a closed subset of X. and let
G = {v 6 F:x ~/- y for some x 6 F}.
which we call the total transform of F by f. It is apparent that G = q(p~]{F)). If
X is a complete algebraic variety, then q: Ty- —> F is a proper morphism and G is a
closed subset of F. Conversely, if G is a closed subset of Y. we call F := p{q~] (G))
the total transform of G by /. If F is complete, then F is a closed set.
Theorem 4.29. Let X and Y be algebraic varieties over a field k. and let f: X- ■ ■ —*
Y be a dominant rational mapping. With k(Y) viewed as a mbfield of k(X) through
f*. we have the following assertions:
(1) Suppose x ~ ^ y for x € X and y £ F. Then f is defined at x if and only if
(2) IfX is normal and Y is complete, then f is defined at every point of codimension
13 of X. Hence. codim^-(A' — U0) > 2. where U0 is the domain of definition of f.
Proof. (1) If x ~f y. then x = p{z) and y = p{z) for z 6 Tf. Let &-_ = &rf.:.
@x = &xx. and &y = @y.v Then we have @-_ > &x and &._ > @y as subrings ofk(X). If
x belongs to the domain of definition U0 of f. then p: T —> X is isomorphic near the
point x. whence &- = @x. So. &x > &v. Conversely, suppose @x > &x. We choose an
affine open neighborhood Spec(B) of y so that B is finitely generated over k. Then
<f, = Bq for a prime ideal q of B. Similarly. <9X = Ap for an affine open neighborhood
Spec(A) and a prime ideal p of A. Write B = k[b\ b„]. Since b, 6 <9X, we may
write bj = «,/,v, («,. ,v, 6 A. .v, ^ p). By replacing A by ^4[1/ n?=i */]• we may assume
that B <z A. Then p n B = q. So the rational mapping f is represented by the
morphism f0: Spec{A) —> Spec(B) ^^ F associated with the inclusion B ^ A, and
fo{x) - y. Thus. / is defined at the point x.
(2) Set Z = Tf. Then p: Z —> X is a morphism which is a birational mapping.
Let x be a point of codimension 1 of X. Namely, x is the generic point of an
irreducible closed subscheme {x} which has codimension 1. Since X is normal, the
local ring <9X of x is a discrete valuation ring of k{X). Then & := &x D k(Y) is a
discrete valuation ring of/:(F). Since F is complete, there exists by Lemma 4.17 a
point y of F such that & > &,.. Then <fv > <f, clearly. Hence / is defined at x, and
y = f(x). ' ' D
Let X and F be normal, complete algebraic varieties, and let f: X ■ ■ ■ —> F be
a birational mapping. The mapping f is defined at a point x £ X of codimension 1.
The point x is the generic point of an irreducible reduced closed subscheme F = {x}.
and the point y = f{x) is also the generic point of an irreducible reduced closed
subscheme Go = {y}. We call G0 the proper transform of F by f. Go is a subset of
the total transform G of F. but G0 is not always equal to G. We can argue similarly
with /-': F • • • —> X and a point y e F of codimension 1. If we put x = f~[(y).
then {x} is the proper transform of {y} by /-1. But we often call {x} the proper
transform of {y} by f. Though it may appear that we use f and f~x confusedly,
in fact, we regard f: X ■ ■ ■ —► F as a correspondence relation (without orientations)
'x is a point of codimension 1 of X if {.\} has codimension 1 in X.

102
II. SCHEMES AND ALGEBRAIC VARIETIES
between the sets of points of X and Y. Moreover, when we speak of the proper
transform G0 of F. we mostly consider the case codimy G0 = 1.
Finally, we state a result concerning the dimension of an algebraic scheme defined
over a field k.
Theorem 4.30. Let X and Y be irreducible algebraic schemes over k. Then we
have the following:
(1) For a closed point x of X. we have
dimX = K-dim(^V.v) = tr. degA. k(XKd).
(2) For a dominant morphism f: X —> Y and a closed point y of Y, we have
dimXv > e. where e = dimZ — dim Y.
(3) If f is a flat dominant morphism in the assertion (2) above, then we have
dimXv = e.
Proof. (1) Since dim X = dim Xred. we may assume that X = Xred. Then X is an
irreducible, reduced algebraic scheme. Let /¾ ^ F\ C . ■. C pn be an ascending chain
of irreducible closed subsets, where n = dimZ. Then Fq consists of a single point
-. and the generic points of the irreducible closed subsets containing - correspond
bijectively to the prime ideals of&x.z (Lemma 4.2 (6)). Hence, K-dim(&x.z) = n. By
problem 1.1.9, we have K-dim(^.-) = tr. degA. Q{&x-)- (Note that Qf@x.z) = k{X).)
Now let x be any closed point of X. Then K-dim(<^V.v) = tr. deg,. k(X) in the same
fashion as above by problem 1.1.9. So we have dimX = K-dim(^y..v).
(2) Let x be any closed point of Xy. By problem 1.1.9. we have
K-dim(^.v) < K-dim(^y.,.) + K-dim(^.Y).
By (1) above, we have
K-dim(^Af.v) = dimJf. K-dim(^y,) = dim Y.
So we have K-dim(<fx,.v) > <■'. Since dimZ, = max{K-dim(<fj-i..v); x e Xv}, we
have dim Xv > e.
(3) By the hypothesis. &x.x is a flat ^y,.-module for any closed point x of Xv.
Since K-dim(<fx|V) = e by problem 1.1.10. we have dimZ,. = e. D
We state the following result without proof.
Theorem 4.31. Let f : X -+ Y be a dominant morphism between irreducible
algebraic schemes defined over a field k. and let e = dimX — dim Y. Then for any
integer r > e, Z := {y 6 Y; dimX,. > r} is a closed subset of Y.
Summarizing the statement of this theorem, we say that the dimension of fibers
of f is upper semicontinuous.
II.4. Problems.
1. The fiber product X xs Y of schemes does not necessarily coincide (as sets)
with the set {{x.y);x 6 X.y 6 Y}. Explain this by making use of the following
example:
Let K be a field, let L/K be a finite, separable, algebraic extension, and let
K'/K be a decomposition field of L/K. Put X = Spec(L). 5 = Spec(K). and
Y = Spec(K'). Then X xs Y = Spec(L ®K K') = TJSpec(/t') (a direct sum of
[L : K] copies of Spec(AT'j).

4. SCHEMES AND ALGEBRAIC VARIETIES
103
2. Let t be a coordinate of the affine line A\. let P be the point t = 0. and let
C/ = A|. — {P}. Let Ibea local ringed space which is obtained by patching two
copies of A[ along the open set U. Then X is a scheme. Show that Ax//,-(X) =
{(Q. Q); Q 6 U}U {{P. P). (P'. P')} mdAx/k(X) = *x,k(X)U {(*"■ P). (/>. /")}•
So X is not separated over Spec(k).
3. Prove the following two assertions on closed immersions relative to the proof of
Lemma 4.9.
(1) Let /: X —> 7 be a closed immersion, and let /': X' —> 7' be the base
change of / by a morphism g: Y' —* Y. Then /' is a closed immersion as well.
(2) If f: X -+ F and g: Y —> Z are closed immersions, then the composite
morphism g • f: X —> Z is a closed immersion.
4. Let /: be a field of characteristic /? > 0.
(1) Let n be a positive integer prime to /?. and let a e /: such that a'/"' ^ /: for each
m.\ <m <n. with m \ n. Let C be an affine plane curve Spec k[x. y]/[y" — ax").
Prove that C is irreducible and reduced, but C (¾ £ decomposes into the union
of n lines passing through the origin of A2k.
(2) If n = p > 0. in the example (1) above, then show that C <% k is a nonreduced
scheme such that (C1¾ £)recj is a single line, though C is irreducible and reduced.
Namely. C <%- £ is the line y = a]l"x counted «-times.
5. Let k be an algebraically closed field. Let Y be the scheme obtained by patching
two affine lines A[ ~ Spec k[t] and Aj = Spec/:[w] together along the open sets
U0 = D{t) and U\ = D{u) by t = irx. Verify the following assertions.
(1) Y is an algebraic k-scheme whose function field is /:(0-
(2) A discrete valuation ring of k{t) containing k is equal to either k[t\,_n)
{a ek) or k[u](u) (cf. (1.1.40)).
(3) Y is separated over k. (Use Theorem 4.15.)
(4) Y is a complete algebraic curve. (Compare Definition 4.17.)
(5) The scheme X in problem 2 above is not complete. The scheme Y is the
projective line P],. which we define in the next chapter.
6. An algebraic variety of dimension 1 is called an algebraic curve. Let X be a
normal algebraic curve defined over a field k. let Y be a complete algebraic
variety, and let /: X ■ ■ ■ —> Y be a rational mapping. Prove that / is then a
morphism. (Suppose / is represented by a pair of an open set U of X and a
morphismg: U —► Y. Consider Z = g(U). If Z is a closed point of Y. then / is
a constant morphism. If Z is not a closed point, then Z is a complete algebraic
curve if Z is provided with the irreducible reduced closed subscheme structure.
So replacing Y by Z. we may assume that / is dominant. Use Theorem 4.29.)
7. Let Ibea normal complete algebraic curve. Show that there is a one-to-one
correspondence between the set of closed points of X and the set of discrete
valuation rings of k(X) containing k. The correspondence is to assign the local
ring &x.s to a closed point x of X. (Use Theorems 4.15 and 4.17.)
8. Let j : X —> S be a finitely generated morphism between two affine schemes X
and S. Prove that Y(X.@X) is a finitely generated r(5,.^?.s)-algebra.
9. Let X and Y be separated algebraic schemes defined over a field /:. let /: X —> Y
be a /:-morphism. and let & be a quasicoherent @x-Module. Following the next
steps, prove that f ^ is a quasicoherent &y-Module.
(1) Replacing Y and X by an affine open set V and f~\V). respectively, we
may assume that Y is an affine scheme. (We shall make this assumption.)

104
11. SCHEMES AND ALGEBRAIC VARIETIES
(2) If X is an affine algebraic scheme, then f ^ is quasicoherent.
(3) Let U and U' be affine open sets of X. If U n U' ^ 0. then U n U' is an
affine open set.
(4) Let {£//}/e/ be a finite affine open covering and set t/(/ = £/,■ n £//. By
(2) above. {f\u,)*{3r\ul) and (f\u,!)*{3r\u,l) are quasicoherent <fy-Modules.
Moreover, we have the natural exact sequence
o-/^- Yiifivxmu.) =t X\{f\vn)mun)-
i i.j
(5) /*^" is a quasicoherent ^--Module.

CHAPTER 5
Projective Schemes
and Projective Algebraic Varieties
In this chapter we define projective algebraic varieties over a field k that play
central roles in algebraic geometry, and we state and prove results relevant to them.
A commutative ring A = Yln<sz -^» *s a 8raded ring if A is a direct sum of abelian
groups An and A„ • A,„ C A„+m (n.m 6 Z) with respect to the multiplication. We
call elements of A„ homogeneous elements of degree n and denote the degree of an
element a 6 A„ by n = dega. The unit element 1 of A belongs to Aq. and A0 is a
subring of A. An .4-module M = ^Z„eZ M„ is a graded A-module if M is a direct
sum of abelian groups M„ and An • Mm C Mn+m for each n.m £ Z. An ideal I of A
is called a homogeneous ideal if I = J2„ez ^« witn I" = I Hv4„. (See problem II.5.1.)
Hereafter, we treat graded rings of various kinds and consider, among others,
those graded rings A — J2„eZA„ with A„ — 0 (n < 0). Then A+ : = YLn>Q^n K an
ideal of A. which we call the irrelevant ideal of A.
Lemma 5.1. Let A = Yln>a^n be a graded ring. Then we have
(1) A is a noetherian ring if and only if Aq is a noetherian ring and A is a finitely
generated A0-algebra.
(2) Suppose A is a noetherian ring. A graded A-module M = Yl„ez M„ is a finitely
generated A-module if and only if the following three conditions are satisfied:
(i) M„ is a finitely generated A^-module for each «£Z.
(ii) for some «o 6 Z, M„ = (0) for each n < no.
(iii) for some tt\. and some d 6 Z, d > 0. M„+ti = A,iM„ (for all n > n\).
Proof. (1) Suppose A is a noetherian ring. Since Aq = A/A+. Aq is then a
noetherian ring. Since A+ is a finitely generated ideal, there exist homogeneous
elements a\ ar such that A+ = Yl\=\^a'- For a homogeneous element b of
degree N in A. we prove by induction on N that b 6 Ao[a\ ar\. Obviously, we
may assume N > 0. Then b = Xw=i c<ai w'tn homogeneous elements r, of degree
N — deg«,. Applying the induction hypothesis to r,'s. we have that c, 6 Ao[a\ ar].
So b 6 Ao[a\ a,]. A is. thus, a finitely generated ^-algebra. The converse follows
from (1.1.14).
(2) Suppose A is generated over Aq by homogeneous elements a\ ar and M
is a finitely generated ^-module. Then M is generated as an A -module by finitely
many homogeneous elements m\ ms. Let c/, = dega, and e, = degm/. Then
any element of M„ is expressed as a linear combination of elements of the form
a"1 • • • afrrrij with coefficients in A0. where n = £V a,^, + e/ and a, > 0. For each
n. there are only finitely many (r + l)-tuples (a\ ar.j) satisfying this condition.
Hence. M„ is a finitely generated ^-module. If we put n0 = min{e7; 1 < j < s}. then
105

106
II. SCHEMES AND ALGEBRAIC VARIETIES
Mn = (0) for each n < «0. Next let d be the least common multiple of d\ dr.
and let b, = («,)''/'''. We let {x/(} stand for the set of all elements a"' ■ ■ ■ a"'mj with
0 < a, < d/dj. 1 < / < r, and 1 < j < ,v. Let n\ be the greatest one of degx/s when
Xfj ranges over all elements of {xM}. If n > nx. then every element of Mn+li is then
expressed as J2< ff(k\ br)xM. So Mn+li = AltMn. Conversely, if the conditions
(i). (ii). and (iii) are satisfied, then we can readily show that M is a finitely generated
^-module. The proof is left to the reader as an exercise. □
In the same way as for the affine scheme Spec(^) for a ring A. we define the
homogeneous spectrum Proj (.4) for a graded ring A = 5Z«>o^» as the set of all
homogeneous prime ideals of A which does not contain A \. For a subset E of A.
define V+(E) = '{<$ e Proj (/!);<£ D E} and D+{E) = Proj (/1) - V+{E). Denote by
E' the set of all homogeneous parts1 of elements of E, and by I the homogeneous
ideal of A generated by E'. Then we have V^(E) = V+(E') = V+{I).
Lemma 5.2. Proj(/4) is given a topology for which the subsets of the form V+{E)
are closed subsets. This topology is called the Zariski topology of Proj (A).
Proof. We have only to verify the following assertions:
(1) K+(0) = ProjU). KfU+) = 0:
(2) V+{Ei)\JV+{E2)= V,(EXE2);
(3) aeA V+{E})= K+(U,eA£,).
The proof of (1)-(3) is left to the reader. □
We shall show next that Proj {A) is a scheme which is obtained as affine schemes
patched together. Let a be a homogeneous element of degree d. In the quotient ring
A[a'x] we define the degree of an element b/a' (b 6 A„. r > 0) to be deg(b/a') =
n—dr. Then ,4[«~']isa graded ring which admits homogeneous elements of negative
degree. The set of all homogeneous elements of degree 0 is the subring A[a~l]0.
Furthermore, if b e Ac then A[(ab)~[]0 = A[a~l]0[(b'' /a'')~x]. In fact, every element
of A[(ab)~x]o is written as x/{ab)k with deg.r = k{d + e). If we choose integers
l.h>0so that k + l = dh. then x/{abf = {xb'/akaeh) ■ {ae/b''f. So A[{ab)'% C
A[a~[]o[(bl1 /a1')_1]. The opposite inclusion is apparent. We shall prove the following.
Lemma 5.3. (1) For a homogeneous element a of degree d. there is a bijection <p
between D+{a) and Spec A[a~l]u which is given by *}} ^> *}L4[tf ~'] n A[a~[\).
(2) If a 6 Aj andb 6 Ae. then D+(ab) corresponds to D{b'' /a') under the bijection
in (1) above.
(3) If we introduce the topology on D+(a) induced by the Zariski topology
on Proj {A). the bijection in (1) gives rise to a homeomorphism between D+{a) and
Specv4[a_1]o. We can therefore regard D+ (a) as a scheme by translating the scheme
structure on Spec A[a~]]o onto D+(a) via this homeomorphism. Then the local ring &
at a point ^3 of D + (a) is equal to {Am)o. where S = {;; - is a homogeneous element
of A and z ¢ *}}} is a multiplkatively closed subset of A and Am is the quotient ring of
A with respect to S. Am is a graded ring admitting homogeneous elements of negative
degree.
(4) Proj(/1) is a scheme separated over Spec(AQ).
1 If we write .y = xr + xr+\ + ■ ■ ■ + ,vs with .v/ £ A j. then each x j is called the homogeneous part of
degree j of .v.

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES 107
Proof. (1) It is apparent that p := *Pv4[a~']nv4[<7~']0 is a prime ideal of A[a~']0.
We shall show that <p{<$) ^ ^(qj') for qj.qy e Z)+(«) with <p ^ <£'. Let p = <p{^)
and p' = pity'). Since *p ^ *p' by hypothesis, there exists a homogeneous element
x such that x 6 *p and x ¢^37. Replacing x by x'' if necessary, we may assume
x 6 Aml. Then x/«" e p and x/a" £ p'. (If x/a" e p'. then x/«" = y/a'" for
>■ 6 Amd n «p'. Hence. av tmx = as+"y 6 <£' (,v > 0). This implies x 6 <£'. which is a
contradiction.) So p ^= p'. Next we shall show that ip is surjective. Let p be a prime
ideal of A[a~%. For each « > 0. let <p„ = {x e A„\xd /a" 6 p}. and for « = 0. let
*Po = {* 6 ^o; x/1 6 p}. We claim that *}} = 52«>o ^P« 's a homogeneous prime ideal
in D+(a) and p = <fi{^P)- By definition. *p is a homogeneous ideal. In order to show
that *p is a prime ideal, it suffices to verify the condition: xy e $. x ^ $ implies
y 6 ^}. Since ^} is a homogeneous ideal, we may assume that x 6 A„. y 6 Am. By
definition, (xy)'1 /a"+m 6 p and x'1 /a" ¢ p. (By convention. «° = 1.) Since p is a
prime ideal, we have y'1 /a'" 6 p. Hence, y 6 tym C ^3. Meanwhile, every element of
p is written as x/a" with x 6 ,4„,/. and x/a" 6 p if and only if x']/a"d 6 p. Then
x 6 <£„,,. and x/a" 6 tyA[a~[] n .4[a_1]o. So p = <?0P). Thus. <^> is a bijection.
(2) We have D+{ab) = D+{a) C\D{{b), and for <# e D+{ab).
¥A[(ab)-']nA[(ab)-% = %A[(ab)-l]n A[a~%[(b'< /a')-']
where p = tp{ty). Hence, p belongs to the open set D(bd/ac) of SpeCv4[a~']0.
Conversely, if p = y>(*p) e D{b''/a"), then a''/«'' £ p and ft''/a'' ¢ p. So by
the definition of *p. we have aft ¢ *p. Thus, y; induces a bijection between D+{ab)
and D{b''/ae).
(3) Note that {D+(aft); ft is a homogeneous element of ^4} is an open basis
for the Zariski topology on D+{a) and that an open set D(x/«")(degx = nd) of
Specv4[a_l]o corresponds to D+{ax) under <p. So ip is a homeomorphism by (2).
Thus, the local ringed space (D+(a). ip*A[u~^]q) is isomorphic to the affine scheme
Specv4[a_l]o. The local ring at the point ty is equal to (^[a_1]0)p. where p = <p{^),
which is nothing but {A<p)o- It can be verified by a straightforward computation.
(4) Let X = Pro)(A). Then {D+{a)}a(zA, is an affine open covering of X.
The coordinate ring ,4[a~']0 of D+(a) is viewed as an ^0-algebra in the natural
fashion. For a homogeneous element ft e A + . denote by oah.a the natural ring
homomorphism v4[a~']0 —> A[(ab)~[]0. Then A[(ab)~x]Q is generated over A0 by
the images crahM{A[a~%) and aah.h{A[b-%). So A[a~% ®A„ A[b-l]0 -> A[{ab)-%.
x (8>j i—> aaf,.a{x) • (Jah.hiy)- is a surjective ring homomorphism. This gives rise to
a closed immersion Au.fc: D+(ab) —> D+{a) xA(> D+{b). Since D+{ab) = D^ (a) n
D+{b). Aa.b turns out to be the restriction of the diagonal morphism A of X onto
D+{ab). So we know that A is a closed immersion. D
Note that the closed set V+(I) of Proj(v4) is identified with Proj{A/I) as sets.
We summarize the properties of Proj(v4) in the following theorem.
Theorem 5.4. For a graded ring A = 5Z«>o^"- ^et % ~ Proj(v4). and let S =
Spec(^o)- Let n: X —> S be the structure morphism with X viewed naturally as an
S-scheme. Then the following assertions hold:
(1) Let 91 be the nilradical of A. and let 91+ = A+ n 91. Then 91 and 91 + are
homogeneous Ideals and Proj ^4/91+ = XK<±.

108
II. SCHEMES AND ALGEBRAIC VARIETIES
(2) Suppose 91+ = (0). Then X is irreducible if and only ifA+ has no zero divisors
in A + .
(3) If A is a noetherian ring, then X is a noetherian scheme and n: X —> 5 is a
finitely generated morphism.
(4) Suppose A is a noetherian ring and is generated by A \ as an A^-algebra. Then
the structure morphism it: X —> S is a proper morphism.
Proof. (1) By problem II.5.1. 91 and 91 + are homogeneous ideals. Let A =
A/91+. Then A is a graded ring. Let p: A —> A be the residue homomorphism.
We have p{A„) = A„. For a homogeneous element a of degree d > 0. define a
homomorphism pu: ^[a_1]0 —> ^4[a~']o by pa(x/ar) = x/ar, where a = p(a) and
x = p{x). We claim that ^4[«_1]o = A[o~x]q/N provided a ¢ 91+. where N is
the nilradical of v4[a_1]0. Suppose x/ar £ A[a~[]o is a nilpotent element. Then
{x/a')' = 0 (for some t > 0). Hence, axx' = 0 for an integer s > 0. So axx 6 91+ and
pu{x/a') = asx/ds+r = 0. Namely. N C Kerpu. Conversely, suppose pa{x/ar) = 0.
Then asx G 91+ for some s > 0. So {axx)' = 0. Then (x/a')' = (<rvx)7(a''4s)' =
0. Thus. Ker^„ C N. The claim is proved. In other words. "(p„) induces an
isomorphism between Spec4[a~']o — D+{a) and (Specv4[a"']o)red = £)+(«)red-
Furthermore, we can show that a{pah) = "(^«)Id,(jm f°r a homogeneous element
b of degree e > 0. Hence, patching {c'{pa)}a&At together, we obtain an isomorphism
between Proj(^) and XK<j.
(2) To prove the "only if" part, suppose there are homogeneous elements a.b
of A+ such that ab = 0. a ^ 0. and b ^ 0. If D+{a) = Spec^[a~']0 = 0. then
^[a~']o is the zero ring, where 1 = 0. Hence, a is nilpotent, and this contradicts the
hypothesis 91+ = (0). SoZ>+(a) ^0. D+(b) ^ 0. and D+{a) nD+{b) = D+(ab) = 0.
This contradicts the assumption that X is irreducible.
The "if" part. If a ranges over homogeneous elements of A + . {D+(a)} is an
affine open covering of X. If we show that all D+{a) are irreducible and X is
connected, then X is irreducible. Moreover. D+(a) is irreducible if v4[«~']0 is an
integral domain. Suppose (x/ar) • (y/as) = 0. Since x/ar = 0 if and only if
a"x/ar+" = 0 (n > 0). we may assume that x.y 6 A + . Then amxy = 0. Since
A+ has no zero divisors, either x = 0 or y = 0. So x/ar = 0 or y /as = 0. We
shall next show that X is connected. Suppose X — X\ \\Xi. Since X\ and A? are
open sets and since {D+(a)} is an open basis of X. there exist nonzero homogeneous
elements a.b such that D+(a) C X\ and D+{b) C X-±- Let d = dega and e = degb.
We have b'1/ac £ 0 in A[a~%. So D+{ab) = D{bd/ae) ^= 0. This contradicts the
assumption that D+(a) n D+(b) = 0.
(3) For a positive integer ¢/, define a graded subring Ai-'r'1 of ^4 by A1-'1^ =
YlnX)^'!"- f°r which the homogeneous part of degree n in A^ is A,i„. By virtue of
Lemma 5.5 below, we have v4[a~']o — A1-'1^/{a — 1) for a homogeneous element a of
degree d. Suppose A is a noetherian ring. Then by Lemma 5.1. A0 is a noetherian
ring and A is a finitely generated ^o-algebra. We may assume that A is generated over
Ao by homogeneous elements a\ ar in A+. Let dj = dega,. In order to show
that X = Proj(v4) is a noetherian scheme and X is finitely generated over S. it suffices
to show that ^[«~']0 is a finitely generated ylo-algebra if a is a homogeneous element
of degree d>0. Let E = {a"1 • • • arQ-: 0 < a, < d. £'_, a-,di = 0 (mod d)}. Then E
is a finite set consisting of homogeneous elements of A^K Note that each element of
(^(''')„ = Al!n is expressed as ]T ea\' ■ ■ ■ a'/ (c 6 v4o- Pi > 0. 5Z/=i /½ = nd)- where
a{' • • • «,-' is written as a product of elements in E. Thus. E generates the v4o-algebra

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES 109
A(d\ Hence. ^[«_']o = A{,])/{a — 1) is a finitely generated ^o-algebra.
(4) Since n: X ~+ S is a separated, finitely generated morphism. it suffices to
show that n is a universally closed morphism. Namely, we have only to show that
the base change n7 : X x.sT -+ T is a closed morphism for an arbitrary morphism
of schemes T -+ S. Replacing T by an affine open set. we may assume that T is an
affine scheme Spec(fi). Then since X = \jD+(a). the construction of X x.s T implies
that X x.v T = Proj(^ ®A{I B). where A ®Al) B = J2n>oA» ®a{I b A closed set of
X x.s- T is of the form V+(I) = Proj(^ (3Al) B/I). where A <$>Alf Bjl is a graded ring
which is generated by A\ ®An Bjl n (A\ (g>A(t B) over B/I n B. It suffices therefore
to prove that the image n{X) of the structure morphism n: X -+ S is a closed set
under the assumption that Aq is not necessarily noetherian. A \ is a finitely generated
^-module and A is a graded ring which is generated by A\ over A(). For a point
y of S. let k(y) = <f,/mr be the residue field. Then n~x{y) = Proj(.4 (g)Alj k(y)).
and >' ¢ n(X) if and only if n~x (>•) = 0. Let ci\ a,- be a set of generators of the
^o-module A\. and let «, be the image of «, in A\ ®Aa k{y). We have the equivalent
conditions:
n~\y) = 0« k(y)[a\/ai «,./«,-] = (0). 1 < / < r.
O for some N > 0. a? = 0. 1 < / < r;
^/4,,8..),, ^(j) = (0). « »0:
<^ A„ ®Aa@y = (0). « >0.
where, in order to show the last equivalence, we use the hypothesis that A„ iX,.j(l <9X is
a finitely generated ^, -module. (Compare this with the proof of Nakayama's Lemma
(1.1.28).] Let /„ be the annihilator ideal of a finitely generated ^-module a„. i.e..
/„ := {c eAQ:c.\ = 0. for each a- eA,,}. Then A„ ®A„ <9y = (0) if and only if y ¢ V(I„).
In fact, write A„ = A{)b\ + • • • + A0b, and &v = {AQ)P. If A„ ®Aa if, = (0). then
for some .v ¢ p such that sbj = 0 for each j. Hence, .v £ /„ — p. i.e.. y ¢ V(I„).
Conversely, if v ¢ V(I„). then there exists some .v £ /„ — p. whence. A„ 8.^,, <9X = (0).
On the other hand. /„ C l„+\ because A\ -A,, = A„+\. Let I = IJ„>o^"- Then I is
an ideal of Aq. and V(I) = f,n>() V(I„), Summarizing the arguments, we know that
y <£n{X) if and only if y ¢ V{1). Hence. n{X) = V{I). So n{X) is a closed subset
of S.
Consider, in particular, the case Aq is a field k. Then n: X -+ S is an algebraic
scheme over S = Spec(^r). In order to show that n is a proper morphism. we may
replace X by X,eii and then Xreii by its irreducible component. So we may assume that
X is irreducible and reduced. There is no loss of generality through these substitutions
nor by assuming that X is given as X = Proj(^). where A is a finitely generated,
graded /c-algebra which is generated by A \ (cf (1) of this theorem and problem II.5.2).
By Theorem 4.17. we have only to show that given a discrete valuation ring {@.m) of
the function field k{X) of X there exists a point x' of X such that & > <fX'. (To verify
that the condition in (1) of Definition 4.16 is satisfied by X. we have to consider an
irreducible and reduced closed subscheme {.x} and a discrete valuation ring k(x) (~\®
of the function field k(x). However, since {.r} is again of the form Proj(^) we may
assume from the beginning that X = {a}.) Let {a\ </,} be a A:-basis of the k-
vector space A\. Then X = (J'._, Z>( («,■). and /)+(«,) = Spec A: [a i/a,- «,/«,].
Let v be the valuation associated with (9. Suppose that min{?'(«,/«i): 1 < / < /•} is
attained by v(a,/a\). Since v(a:/a,) = v(a,/ct\) — v(a,/ci\) > 0 (for each i. 1 <i<r).
we have k[a\/a, «,/«,] ^ <9. Let p = m n k[a\/a, «,/«,]. and let x' be the

10
II. SCHEMES AND ALGEBRAIC VARIETIES
point of Z>4 («,) corresponding to p. Then (9 > k[tt\/a, ar/a,]p = &x,. Hence.
X is proper over k. By Lemma 4.19. we conclude that the structure morphism n is
a proper morphism. □
In the above proof, we made use of the following lemma.
Lemma 5.5. Let A = J2„>o^" ^e a Sraded ring, and let A1'-'1' = X3„>()/L/„ for a
positive integer d. Then for a homogeneous element a of degree d. we have
A[a-l]o*AW/{a-\).
Proof. Define a ring homomorphism p: A^'1^ /(a - 1) —► ^[«~']o by x £ Adn i->
x/a". Evidently, p is surjective. We shall show that p is injective. Suppose p {x\ +■■ ■ +
x,) = 0(.t, £Alhli). Forn >n, (for each i). p{x\ + \-xx) = x]/a"< H + x,/a'h =
{a"-"'x{ H +'«"-"-x,.)/«". So a'{an-"K\\ + h a"-"'xs) = 0 for some I > 0.
Hence, .v, + • ■ ■ + xs = (x, - al+"-'"x\) + ■■■ + (x, - a'+"-'uxs) £ (a - l)v4(,/>. D
Definition 5.6. When A is a noetherian graded ring which is generated by A\
as an ^o-algebra. then X = Pro j (,4) is called a projective scheme over S = Spec(^o)-
The structure morphism is then a proper morphism (Theorem 5.4 (4)). When Aq
is a field k and X = Proj(A) is an algebraic variety, then X is called a projective
algebraic variety.
A basic example of projective scheme is given by the polynomial ring A =
R[T,). T\ T„] in n + 1 variables over a ring R which we view as a graded ring
by putting A{) = R and deg T, = 1 (0 < i < n). Then Proj(^) = (JLo^M7/) and
Z)4 {Tj) = Spec/?[T0/T, T„/Tj] = A"R (the affine space of dimension n over R).
If ;' 7^ j. then T)+(T,) and D+{Tj) are patched up in Proj(^) along the respective
open sets D{Tj/Tj) and D(Tj/T,) by the isomorphism which is associated with an
R[Tj/Tr Tj IT,]-algebra isomorphism
R[T{)/T, T„/Tr Tj/T,\ -> R[T0/T Tn/Th T,/T,]
Ti/T, ~ T,/T, = T,/Tt ■ Tj/Ti (I ? i.j).
We then denote Proj(^) by P"R and call it the projective space of dimension n. We call
(T().T\ T„) its homogeneous coordinates. If R is a field. P'[. is an algebraic variety
over k. We shall consider the set of closed points P"(k) when k is an algebraically
closed field. Let P £ P"(k). Then P £ D+{Tj) for some i. In order to simplify the
situation we assume i = 0 in the subsequent argument. Let t, = T-,/'7b '(1 < i < n).
Then P corresponds to a maximal ideal m = (t\ — a\ t„ — a„) (a, £ k). So. the
homogeneous prime ideal ^3 which corresponds to P is *}3 = {T\ — a\Ta T„ —
a„T0). This is considered as the defining ideal of a line (Tq.ohTq a„T0) on
A/'+1 = Spec£[7o T„] which passes through the origin (0 0). If q, ^ 0. then
the line is written also as ((l/a,-)To. (ai/a,-)Ti {an/a;)T„). Conversely, given
a line (/?()T0./?iTo (i, T0) (/?, £ k) passing through the origin, we have /?, ^ 0
for some i. Then the line defines a point (/?o//?,- Pi-\/Pi-Pi+\/Pi Pn/P<) of
T)+ {Tj). Namely, there is a one-to-one correspondence between P"(k) and the set of
lines on A^+l passing through the origin. A line on A£+l passing through the origin
is determined by its slope (/¾./?i /¾). and the slope is uniquely determined up

5. PRO.IlXTIVi; SCHKMhS AND RROJ1 C'TIVI ALGliBRAIC VARM/llhS 111
to the equivalence relation (/¾./?i /?„) ~ (//¾. Mh //") U S k* := k — (0)).
Thereby, we obtain a one-to-one correspondence
P"{k) <-»*-"-' -(0)/-.
where (ao a„) ~ (/¾ fi„) if and only if [i, = ka, (for each i) for some / £ A".
To this effect, we denote a point (a„ a.,,) of P"(Ar) by (a(l : a, : • ■ ■ : a„).
Let / be a homogeneous ideal of A = k[T{) T„]. Then K, (I) is a closed set of
P" and the closed subsets of P". are exhausted by those of the form V, (I). The residue
ring B = A/I is a graded ring which is generated by B\ = A\/A\ n / over Bo(= A).
Since every homogeneous prime ideal of £ is expressed as ^3// with *}3 e V, (/).
the topological space K+(/) is identified with Proj(fi). Let X = Proj(fi). X is then
an algebraic scheme which is separated over k. By Theorem 5.4. X is reduced if
I = \J1 and an algebraic variety over k if / is a prime ideal. (The hypothesis that
k is algebraically closed is still in force.) As a homogeneous ideal of the polynomial
ring A:[To T„\. I is generated by finitely many homogeneous polynomials
f, =//(75) Tn). 1 <j <N.
If P = (a, a„) is a point of P"(Ar) and ^3 is the corresponding homogeneous
prime ideal of A. then we have the equivalence
P e X(k) iff <Pd/ iff /,(00 a„) = 0. \<j<N.
The first equivalence is apparent. We shall show the second equivalence. For the sake
of simplicity, we assume a() ^ 0. Then P £ /)^(7,)). and X C~\ D t (70) is the closed
subset in Spec k[t\ t„] defined by the ideal /() of k[t\ t„] which is generated
by {/,(l.fi t„); 1 <./ < N}. Hence, if P € XnD+{Tn). then
(*) fiU-~ -)=^^/,(^0.0, a„)=0. \<j<N.
V ao on) J
Thus, we know that the "only if" part holds. Conversely, if (*) holds, then P e
X n /)+(7()). whence the the "if" part follows.
Since an irreducible reduced algebraic scheme defined over an algebraically closed
field is an algebraic variety. X is therefore an algebraic variety. As we explained
after Lemma 4.11. we confuse (X.<fx) and {X{k).ifx{k)) and call X{k) a projective
algebraic variety defined by a system of equations //(To T„) = 0 (1 < j < N).
When the ideal I is generated by a single homogeneous polynomial / =
/(T). T, T„). we call X a hypersurface defined by / = 0 without caring whether
or not X is reduced (or irreducible). We call deg / the degree of the hypersurface
X. In particular, if/ is a linear polynomial
/ = «„T„ + «, T, + • ■ ■ + a„T„. («o «„) e A"H - (0).
then we call X a hyperplane. Suppose the linear polynomials f = ^" ()«/T, and
# = Yl"i o^iTf define the hyperplanes //(/) and H(g). respectively. Then we have
//(/) = //(#) if and only if for some k e k*. b, = ka,. 0 < i < n.
We shall show "only if" part. For the sake of simplicity, assume that //(/) D
D.(T()) = H{g) n/),(7,,) 7^0. Then//(/) n7), (T„) is defined on Speck[t\..... t„]
by a principal ideal (ao + a\t\ + \-ant„) and H(g) nD, (T0) by (/¾ +/7^1 + h
A„?„). So the two ideals (w() + «i?i + • ■ ■ + «„/„) and (fc0 + h\t\ + ■ ■ ■ + h„t„) coincide

II. SCHEMES AND ALGEBRAIC VARIF.TIES
with each other, which occurs if and only if for some 1 6 k*. bo + b\t\ + ■ ■ ■ + b„t„ =
/.(wo + ci\t\ + ■ • • + a„t„). We have thus shown the "only if" part. The converse is
apparent. Consequently, the set of hyperplanes on P';' is in one-to-one correspondence
with the set of k-rational points of another projective space (P)')'- the correspondence
being the assignment Yl"-oai^ = 0 ^ ("<> c'n)- In the case where k is an
algebraically closed field, we call (P".)' the dual projective space of P';' in the above
sense. The dual projective space of (PJJ)' is the P"k which we started with.
The previous observations are quite similar to the ones we made on affine schemes
Spec(^?) corresponding to rings R. We constructed morphisms of affine schemes,
which correspond to ring homomorphisms R —► S. and Modules M corresponding
to ^-module M. We shall observe below that similar objects are constructed for
homogeneous spectrums as well. Whenever we consider a graded ring A = ^2,, >() A„.
we assume that A is generated by A\ as an ^o-algebra.
Let A = J2„>o^" ar,d & = H„>o-^" be graded rings. A ring homomorphism
ip: A —► B is said to be a homomorphism of degree 0 if tp{A„) C B„ for each n > 0.
For a homogeneous prime ideal Q of B. ip~x (£3) is then a homogeneous prime ideal.
Note that ^"'(Q) 2 /j+ if and only if O. 2 <p(A ,). Let G{ip) = D+(tp(A~)).
Then we can define a mapping of sets f: (?((/?) —► Proj(^) by £J >—» y ~' (£3). Since
/~'(D.t(tf)) = D^{ip{a)) C (?((/?) for a homogeneous element « of ^ + . / is a
continuous mapping if the open set G(ip) is given the induced topology on Proj(fi).
We claim that / is a morphism of schemes. For a homogeneous element a e
A ,. we can define the natural ring homomorphism v?(«): A[a~']o —► 5[/>_l]o by
(/?(„)(x/a") = ip(x)/b" for a-/«" e ^[«_,]0. where A = ¥?(«)■ This induces a
morphism "(<£(„)): D( (fc) —> Z)4 («). where we note D+{b) C (?((/?). If a ranges over
all homogeneous elements of /lf. {£>+(cp(t/))} is an affine open covering of G(tp).
For homogeneous elements a. a' of A + . the restrictions of "(v(„)) and "(V(</')) onto
D+(ip(aa')) = Da (</?(«)) H /)+((/?(</')) are identical. So in view of Lemma 4.3. we
get a morphism of schemes "tp: G(y) —> Proj(^). which coincides with / as the
mappings of sets. So f is a morphism of schemes. By the above construction, we
know that "ip is an affine morphism (cf Definition 4.20).
As an example, take A = k[Tn T„]. For a matrix P = (/>,,) belonging to the
general linear group GL(« + \.k). define an isomorphism ipP: A —> A of degree 0 by
ipr(Tj) = J2"j uPijTj- If we note that ipP(A + ) = A + . we see that G(ipP) = Proj(^).
Let //> = "(ipp). Then /p is a ^-automorphism of P)'. and f p- f q = f Pq. Moreover,
if aP denotes a scalar product with a e k*. then we have f /P = fP. Meanwhile,
the center of GL(« + \.k) is {aI„+\: a £ k*} = k*. where /„fi is the identity matrix
of size n + 1. Let PGL(n.k) = GL(« + \.k)/k*. Then the mapping P ^ fP is a
group homomorphism /: PGL(n./c) —> Aut(P'/'). where Aut(P'A') is the group of all
Ar-automorphisms of P'j.. the multiplication being the composite of morphisms. In
fact, it is known that / induces an isomorphism of groups PGL(«. k) = Aut(P'/'). We
call f p the projective transformation on P';' defined by P and PGL(«. k) the projective
transformation group in dimension n.
Let M = J2„e'/. ^" ^e a 8ra<Ied ^-module. For a homogeneous element a of
A ,. define an ^[w'Lmodule M[a~]] by M[a~l] = M £5 A A[a~[\. Then M[a~]] is a
graded module. If x e M„ and a e A,/, then deg(.v/«"') = n —dm. The set M[a~l]0
of elements of degree 0 is an A[a~ 'jo-module. So we can define a quasicoherent sheaf
M[a-%\ For a e A,, and b e ^,. M[(«A)-']~ is identified with (M^/"1],)^''//)''])^
if we identify Z>( (ab) = D^ {a) n£>, (A) with the open set D{bd /ac) of Spec/l[«_1]o.

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES II?
For example, an element x/a'b"' e M[(ab)~]]Q is written as x/a'b"' — (x'/a1'''/') •
(a1'/b'1)!1 which belongs to M[a~]]o[a1'/b'1]. where we choose ft so that d(l > in and
put x' = /7''/'~"'.y. Namely. M[(ab)~x\) C M[a~]]{)[aL'/b'1]. The opposite inclusion
is apparent. Hence, if a ranges over all homogeneous elements of A^. then the
{M[«"'](7} get patched together to give a quasicoherent Module on Pro)(A). which
we denote by M.
Let /: M —> N be a degree 0 homomorphism of graded ^-modules, i.e.. /(M„) C
/V„ foreachn. Fore/ €^,/. we can definea homomorphism /,7 : M[« ~'],7 —> /V[« ~'](7
by /«{x/a") = f(x)/a" and patch {/,7} together to give a homomorphism /: M —>
/V of quasicoherent Modules. We shall observe in Theorem 5.9 that a quasicoherent
Module on Proj(^) is written as M.
Given graded ^-modules M and N. we define a grading in M <g>A N by deg(.v ¢0
y) = p + a. (x G Mp. y e Nq) which makes M <E)A N a graded ^-module. For a g A,/.
define the natural ^[«_1]o-homomorphism
A„ : M[a'% ®A[l,-% N[a~l]0 — (M ®A N)[a~l]0
by J.a((x/ar)<%>{y/ci,s)) = (x <g>y)/a'+A. If« ranges over all homogeneous elements of
^4+. we can patch {A7} together to give a homomorphism of quasicoherent Modules
A: M Sj N ->• (M ®^ JV)~.
An ^-homomorphism /: M —> /V is a degree n homomorphism if /(M,) C M„,,
for each r. Let Hom^(M. /V)„ denote the set of all degree n homomorphisms from
M to /V. and let HomA(M.N) = ^^Hom^M. N)„. Then HomA{M.N) is a
graded ,4-module. For a e Atj. define an ^[tf~']()-homomorphism
Ha\ Hom..,(M. #)[""']<) --■ HomA[a -ih{M[a-l]{). N[a -']<,)
by jua(f/a"){x/a') = f{x)/a"ir (/ e Horn^M. 7V),„/. x e /1,,,). If a ranges over
homogeneous elements of A+. we can patch {«7} together to give a homomorphism
of quasicoherent Modules
/1: Hom.^(M. /V)~ -> <%%^(M. /V).
Lemma 5.7. Lc? ^4 Ac « noetherian graded ring such that A is generated by A\ as
an A{)-algebra. and let X = Proj(^). For graded A-modules M and N. the following
assertions hold:
(1) If M is a finitely generated A-module. then M is a coherent <9x-Module.
(2) A: M (¾ ~ N —> (M (g)A N)~~ is an isomorphism.
(3) If M is a finitely generated A-module. the homomorphism jl: HomA(M. N)~~
—► SPam|(A/. N) is an isomorphism.
Proof. (1) For a e A\. A\_a~x]^ is a noetherian ring (cf. the proof of Theorem
5.4 (3)). Moreover. X is a noetherian scheme and is covered by a finite number of
open sets D+(a) (a e A\). So in order to show that M is coherent, it suffices to
prove that M[a~]]o is a finitely generated ^[«~']0-module. Since a e A\. we have
^["~']o — A/(a - I)A and M[«~']0 = M/(« - \)M by the same argument as in
Lemma 5.5. Then M[«_l]o is a finitely generated /l[«~']o-module because M is a
finitely generated ^-module.
(2) Since X = (Ju£ ^ Z)+ («). we have only to show that A„ is an isomorphism for
each a e A\. Clearly. A is surjective. Suppose A„ (^, (.t,-/«'"') ® (>•//«"')) = 0. Let
m = max,-(m,) and n = max,-(«,■). Replacing .r, and y, by a'"~'"'x, and a"~'hy\. we

114
11. SCIII-.MFS AND ALGKBRAIC VAR1FT1I S
may assume that .v, 6 Mm and v,- t N„. Since (J2, ■>•'/ &.)',■)/""''" = 0. «' ^,- .v, Jo v, =
Y,; a1 xj C< .v, = 0 in M fro, N for some / > 0. If we recall the definition of M (K,t N.
J2, «'(.v,-..v/) is an element of the ,4-submodule R of the free ,4-module F generated
by the elements of M x N. where R is generated by such elements of F as
((11 -f (/[. C|) — ((/1 . l'\) — ((([. V'l). (((2. ('2 + '('2) — {i'i-'<':) — ("2- ''2)-
(biii,. t<i) — (ity-hi'i). and r(w4. (.4) — (('M4. (.4).
where iij.Vj.b.c are homogeneous elements of M. A7. ^. respectively. Namely, we
have a relation: ^V «'(.\7.j',-) = a linear combination of the elements as above in R.
Noting that dega =-- 1. divide both sides of the linear relation by a suitable power
of a. Namely, if deg(/ = ;•. degv = .v. and degb = t. then replace u. v. h by <//«''.
t'/a\ bla'. respectively. Then Y^.,(cl''A'<A''""'''■>'//"") belongs to the subgroup jR0 of
the free /4 [«~'Jo-module /¾ generated by the elements of M[« ~']o x jV[«~'](). such
that F()/Ri) — M[« '](> c*,i[„ i]„ ATw-1])). This implies that ka is injective.
(3) To prove the assertion (3). we need the following definition.
Definition 5.8. Let A be a graded ring, and let I be an integer. Define a graded
^-module A(l) by A(l) = 5Z„e7.^(0n and A(l)„ = A,^„. A(l) is then a free,
graded ^-module with generator 1 e A(l)_/. For a graded ^-module M. define
a graded /1-module M(/) by M{1) = £„eZM(/)„ and M(/)„ = M,+ „. Clearly.
M(/ +in) = M{l){m).
We resume the proof of (3). Since M is a finitely generated ^-module, it is
generated by finitely many homogeneous elements {.y,: 1 < i < «}. Let /, = deg.v,. and let
P = 0" 1 A{—lj) be a free ,4-module. We then define a surjective .4-homomorphism
((: P —> M of degree 0 by l(e .4(-/,)/,) 1—> .y,-. Since ^ is a noetherian ring. Ken/ is a
finitely generated ,4-module. So we find a surjective /1-homomorphism v. Q —> Kerw
of degree 0. where Q = 0' ., A(—mj) is a free ^-module. Thus, we obtain an exact
sequence of graded A -modules
Q A P A M -+ 0.
where P and Q are finitely generated, free graded /1-modules. For a e A\. the exact
sequence yields an exact sequence of v4[a~']0-modules
Q[cr%±P[a-\%M[a-\^Q.
Employing the abbreviations R = .4[«~']o and Pu = P[a~x\). we have a commutative
diagram
0 > Horn,,(M. AT),, —^-* HomA(P.N)a -^-+ HomA(Q.N)u
1"" 1'"" 1""
0 * Hom«(M„.Af„) ---^-+ Hom«(/>„./V„) -^11^ Hom*^,. W„).
where the upper and lower rows are exact sequences. If//,, is an isomorphism for
free /4-modules P and Q. then //„: HomA(M.N)tl -+ HomR(Ma. Na) is an
isomorphism. Here we have HomA(P.N) = 0"_, Horn,, (,4(-/,-). /V) = 0--, A7(/,).
where the isomorphism Hom.j(/4(—/,-). A7) = A7 (/,-) is obtained by assigning to / e
Horn, (A(-I;).N)k the image /(l)(e W,,+A = ^(/,),,) of 1 G .4(-/,-),,. If we show
that //„ is an isomorphism for M = .4(-/). then //„ is an isomorphism for P and Q

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES
15
as well. Since A{-l)a = A{-l)[a~% = R{l/a'). fia: N(l)a -► UomR(A{-l)a. Na)
is given by jUu(x/ak){\/a!) = x/ak+l (.v G Nk{i)- Suppose fiu [x/ak) = 0. Then
a"x = 0 for some .v > 0. and hence, x/ak = a\x/ak+" = 0. So//,, is injective. Given
<p G UomR(A(-l)a.Na). let <p(l/a') = x/ak {x G Nk). Then fil,(x/ak-') = tp. So
fj.u is surjective. D
Let A be a noetherian graded ring such that A is generated by A \ as an ^0-algebra.
and let X = Pro](v4). Put &X{1) = A{1)~ for an integer I. Since A{l)[a~% =
A[a~%(a' ■ 1) for a G A\. we have ^x{l)\Dja) - ^x\o+u,)- Hence. &x(l) is a
locally free ^-Module of rank 1 (cf. (1.2.10)). By Lemma 5.7. we have &X{1) ^¾
&x{m) = (A{l)®AA(m))~ = A{1 + m)~ = cfy(/+m) and JTW, (cfX{1).<fx{m)) =
HomA{A{l).A{m))~ = A{m - /)~ = &x[m - I). In particular, if we denote by
[&x{l)] the isomorphism class of cfx(l) as cfy-Modules. the set {[ffx{l)\.l G Z}
becomes an infinite cyclic group with the multiplication defined by [<fx {1)] • \<9X (m)] =
\@x{l) ®fi\ @X{m)]. We will see later that (fx(l) is an example of invertible sheaf.
Now let T*(&x) = J2„e/.r(x-^x(n)). Then T*{&x) is a graded ring with the
elements oiT{X.@x{n)) viewed as homogeneous elements of degree n. For b G A„.
b/\ e A(n)[a~]]0 = T{D+{a).<fx(n)). where a G A]. Hence, there is the natural
homomorphism of abelian groups (or modules over a field k) a„: A„ -^>T(X.(fx(n)).
b h^ b/\. Then a. = ^„e7c*„: A —> T.t((fx) is a homomorphism of graded rings.
Let 9~ be an &x-Module. Let ^{n) = & ®gx @x{n). and let
r*(^) = ]Tr(x.^(«)).
neZ
Then r*(^") is a graded T,(cf^)-module by the natural mapping
TiX.^in)) ®T{X.@x{m)) -► Y{X.9'{n + m)).
Hence, we can regard r*(^") as a graded ^-module via the homomorphism a. We
then define a homomorphism of cf^-Modules /¾^ : T^tF)"" —> S^" as follows.
For « G Ac, and x G 1^(.^),,,/. we have a-/«" G r*(^)[a-]]0 and (.v|D („))/
(a</(«)|o4(rt))" £ r(Z>+(«).Sr). Define a homomorphism of ^[«_1]0-modules
by a-/«" h-> (A|Dl((,))/(a(/(«)|£1|(„))". For b e Ae. we have (/?r)„,7 = (/?r)„k<„/,)■
Then {(/??-)«} are patched together to give fi& : r»(Sr)~ —> S*".
We have now the following result.
Theorem 5.9. Lef ^ Ac c; noetherian graded ring such that A is generated by A\
as an Ay-algebra, and let X = Proj(^).
(1) If A is an integral domain, then a.: A —> Y„{<^x) is injective. If A\ = J2i-\ ^o«/
and all the a, '.v are prime elements of A. then a is an isomorphism.
(2) Ij'y is a quasicoherent @x-Module, then fi& : T*(Sr)~ —> S*" is an isomorphism
of (9x-Modules.
Proof. (1) Let b G .4,, (n > 0). Ifa„(6) = 0. then a(b)\DAa) = (b/af) ■ a'J = 0
for each i. Hence, b/a" = 0 for each ;'. Since A is an integral domain and A^ is
generated by A\. we have b = 0. So a is injective. We shall prove the rest of (1).
Let .v e r(X.(fx(n)). Then we can express .v|/^(,,,) = {bj/ti"') • a" with fc, G ^,,, with
n, > «. Since (bj/a"') • a" = (bj/a"') • a", on D+{a;cij). we have /?;«"' " = hja"'~".
Since c/, is a prime element. «,-1 b; or a,- | «;- provided n, > n. If a, \ ar «; = ca, with

16
II. SCHEMES AND ALGEBRAIC VARIETIES
c G Ay). Since «, is also a prime element and ctj\c. we have at \ a,. So c G (/4o)* =
{invertible elements of A0}. Thus, we may assume a, \ bh This implies that if «, > n
we may replace b; by A,-/«,-. So we may assume n, = n. Similarly, we may assume
n, = h. If we put h = bj. then we have a.,,(b) = .v.
(2) X = |J' , £>-(«,-) is an affine open covering of X. Since {/)+(«): « is a
homogeneous element of /1^} is an open basis of the topology on A\ in order to prove that
/¾. is an isomorphism, it suffices to show that (fJgr)a: r„(.!?")[«-l]0 —> r(D+(«).Sr)
is an isomorphism. Let a e Ad. Assume that (.v|o4(<,))/(a</(")lz)t(«))" = 0 f°r
seT{X.9'{nd)). Thisimplies (^10,(,,,^/(0,,(^)10.(„Uj))" = 0. Considering s \,h(Ui) e
r(^(a,-).^(/iJ)). we then have (a,,(«)|fl+(„,))'" • (^,(,,,,) = (a,,(a)m.v)|D( m = 0
for some m > 0 (for each /). Hence, considering a,,(«)'" • .v e T(A'.^"((« +
;m)</)). we have a,,(«)'".v = 0. In particular. ,v/a" = q(,(«)"'.v/«"+'" = 0. That
is to say. 0?y)« is injective. Conversely, let f e r(D+(«).Sr). Since f\o,(aa,) e
r(JD+(««,).^)-r(/)+U).^)[(«/^')-'].(/ja)(ao,))-(«ja, („„,))" is extended to an
element .v, £ r (£)+(«,-). SF(nd)) for an integer « > 0. We can choose « s> 0 so that « is
independent of/. Then the restrictions of ,v,|/)t(UU/) and .y,-|p+(„u ) onto D+(aa,a j) are
identical. Hence, there exists an integer am > 0 such that (.v,- • a'" — .v; • a'")\Dt („ „ ) = 0
in T{D^ (ajtij).SF{(n + m)d)). Since we can choose am independently of i. j. we have
shown that / • «"+'" is extended to an element .v of r(X.&~((n + m)d)). It is then
apparent that (/i.y)„(x/a"+l") = f. (Compare the proof of (2) with the proof of
(1.1.12).) □
Corollary 5.10. Let A be the same graded ring as in Theorem 5.9. Let & be
a coherent @x-Module. Then there exists a finitely generated A-module M such that
F ^ M.
Proof. By Theorem 5.9. & ^ N. where N = T^). Write Ax = £^, Am
and X = |J'._! D{(a,). Over an affine open set D+(a,) = Spec4[a,~']0. HF\D^Ui) =
N[a~]]y and N[a~]]0 is. hence, a finitely generated A[a~']0-module. Let {*,-,■/«,'':
1 < / < r,} be its system of generators, where x-,j is a homogeneous element of
degree d in A/", d being independent of /._/. Let M = ^,- .Ax;-r Then M is a
finitely generated, graded ^-submodule of N. Since M[«~']0 = N[o~1]q. we infer
that M = &. D
Hereafter, /c is a field and jR = k[T(, Tn] is a polynomial ring in n + 1 variables.
We consider a graded ring A which is the residue ring of R by a homogeneous ideal
I of R (1 ^ R,). Then A is a noetherian /:-algebra. Aq = k. and ^ is generated by
/fi asa /t-algebra. ^ is an integral domain if and only if I is a homogeneous prime
ideal. Proj(7?) is the projective space of dimension n. which we simply denote by P.
We set X = Proj(/4) as above. A' is a closed subscheme of P defined by the ideal
sheaf/. Denoting I by Sx. we have an exact sequence of ^-Modules.
0-^-^-^-^0.
For an integer in. define invertible sheaves @P(m) and ^(am) as above. Then ^-(am) =
$p(am) (¾¾. <9X. Hence, we have an exact sequence of ^fp-Modules
Q^Sx{m) ->&P{m) -->&x{m) -> 0
which yields an exact sequence of /c-modules
0^r(P.J?x(m)) -> r{P.&e(m)) -> r{X.ffx(m)).

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES
By virtue of Theorem 5.9 (1). we have
0 (m < 0).
k[T0 7-,,1,,, (m>0).
r(P.^P(m)) = |
If m > 0. an element / of T(P. J'xtyi)) is a homogeneous polynomial of degree m
in 7"„ T„ such that f/T!" e T(D+(Ti);^x) = IR[Trl] n R[Tr%. So 7// e /
for r > 0, where we may assume r to be independent of i. Since 7 7j JR + . we have
Tj ¢ 7 for some i. If 7 is a prime ideal (i.e., A is an integral domain), then f € I.
Thus. r(P. J^m)) = /„,. Hence, we know that A,„ •—> Y{X .&x{m)) provided A
is an integral domain. By virtue of the cohomology exact sequence (1.2.16). A,„ =
r(X.@x(m)) provided 7/1 (P.Sx(m)) = (0). _
Let &~ be a coherent ^-Module. Then by Corollary 5.10. !? = M for a finitely
generated graded ^-module M. Since A = R/I. we can view M naturally an R-
module which we denote by MR. Then the ^fp-Module {MR)~ is coherent (Lemma
5.7) and equal to i*y. where (: X <—> P is the natural closed immersion. Since MR
is a finitely generated graded jR-module, we obtain, as in the proof of Lemma 5.7
(3). a surjective homomorphism of ^fp-Modules
0^pK)^ 1,9-^0.
;-I
Let &\ = Ker>7. By (1.2.12). &\ is a coherent ^fp-Module. By the same reasoning
as above, we have an exact sequence of ^p-Modules
/i
0^^^0^(^,,)^^, ^0.
/-1
We can repeat the above argument to construct coherent ffp-Modules ,9^-^- • • •
Namely, we have an exact sequence of ^fp-Modules for j > 1.
/-1
Then the exact sequence of (fp-Modules
- ©<*•(«/,-) - ©^p(«/-i/) ^---^ 0^p(«,,)
/--1 /-1 /-1
/*
^0^(^.)^,^-+0
/-1
corresponds to a resolution of A7« by free, graded jR-modules
/-1 /-1 /-1
Now we utilize the following result.

118
II. SCHEMES AND ALGEBRAIC VARIETIES
Lemma 5.11. Let Ao be a noetherian ring, and let X -+ Spec(^0) he a
separated, finitely generated morphism. Then for a quasicoherent <fx-Module 3^. we have
H"{X.^) = H"(X'.&) for eaeh n > 0. If we have a finite affine open covering
H = {£/,}/, ofX. then H"{X.&-) = (0) for each n > n'.
Proof. For an arbitrary open covering of X. we can choose its refinement which
is an affine open covering. So if U = {£/,-},-e/ is an affine open covering of X.
we have only to verify that H"{U,.!F) = (0) for each n > 0. m > 0. and .v =
(/'o /,„) e /"''' (cf (1.2.22)). Here C/s = £/,-„ n ■ • • n £/,-„ is an affine open set. and
its coordinate ring R, is a finitely generated A0-'d\gebva. (If U and V are affine open
sets. U n V = (AX/S)-](U x.v V) (S = Spec(^„)). and since U xs V is an affine
scheme and AX/s is a closed immersion. U D V is an affine open set of X. From
this we find C/v to be an affine open set. On the other hand, if U is an affine open
set of X. the structure morphism U -+ Spec(^o) is a finitely generated morphism.
By problem II.4.8. T{U.<fc) is then a finitely generated algebra over A0. It follows
that /Jv = T{US.@X) is a noetherian ring.) By means of Theorem 3.14. we conclude
H"(Us.Sr) = (0) for each n > 0. and so. H"{X.&) = H"{X.^) for n > 0.
Let It = {£/,},"_, be a finite affine open covering of X. By (1.2.22). H'^X.^) =
//"(il..?") for each « > 0. For« > N. an element of H"{iX.&') is represented by a Cech
cocycle a which assigns to each .v = (;'o /„) £ I"+l an element a(.v) £ r(C/v. ^").
Since n > N. there is an index appearing repeatedly in .v. So a(.v) = (0) by the
definition of Cech cochains. So //"(11. SO = (0) for each n> N. □
Now going back to P = P". we have an affine open covering 11 = {C/,-}"_0 of P.
where t/,- = Z) + (7,-). We shall prove the following lemma.
Lemma 5.12. For an invertible sheaf (fp(m) (m G Z) on P. the following assertions
hold:
(1) H°(P.&P(m)) = (0)./or each m < 0: //°(P.^p(w)) = Ar[70 7„],„/or ear/i
m > 0.
(2) //'(P.^p(w)) = (0) /or each I. 0 < / < n. and m e Z.
(3) //"(P.^p(w)) = (0)/or m > -(« + 1): //m < -(«+ 1). //"(P.^P(m)) /.v «
k-vector space of dimension ( ~'"~').
Proof. (1) follows from Theorem 5.9 (1) and the definitions.
(2) Let 11 = {£/,-}"_,) be the affine open covering as given before Lemma 5.12.
Then H'(P.<fp(m)) = II'(U.S'p(m)). Following the definition of Cech cohomologies.
we compute //'(il.^p(m)). Set I = {0. 1 «}. For (/0 //) e /'+'. let £/,-„ ,, =
fl'-o £/.-,- Then r(t/,-„ ,-,.^P(m)) = k[T0 7,,. 7,,;1 J);1],,,. A Cech cochain
a e C'(ll. ^p(w)) is viewed as a function which assigns to each (j'o //) £ //+1 an
element a(/(> //)= /,,,...,,/7,- '" ■ • ■ 7) ''. where/,-,...,-, is a homogeneous polynomial
in /c[7() 7„] of degree deg /,-„...,, = m + a,„ + • • • + a,-, (a,-; > 0 for each j). Here
we have /,„...,, = 0 if /,- = ik (for some j.k.j / k) and /^,-,,)...aii/) = sgn 0)/,-,...,-,
for any permutation rr of {0. 1 «}. Since //+1 is a finite set. we may express
a(/0 //) = /,,,...,,/(7,-, • • • 7,-,)' for r > 0 which is independent of (/0 //). Then
a is uniquely determined if /,,,...,, is given for every (;'o //) with /0 < i\ < ■ ■ ■ < //.
Furthermore, a e Z! (!d.<fp{m)){: = Kerof') if and only if we have
/ + i
E(-1)'-A,-/,-^/(^■■■?•; ---^,,/ = 0
/-0

5. PROJECTIVE S("HEMF:S AND PROJECTIVE ALGEBRAIC VARIETIES
for any (/'o < /| < ■ • • < //, i). i.e.. the equality
/ ■ l
c_iv'-'r.'' i. . .
hi---/, ■••/(. i
^/,..,,,=^(-1)^^:./:
/-1
holds. Here ./,-,../..., is a homogeneous polynomial of degree m + r(/ + 1). which
we regard as a polynomial in Tin with coefficients in k[Tu 7/,, T„] and express
.' /11---/,---/,,i /0 /I,---/,---//.i A'd--■/,■ ■■//■ i
deg, £f, - < /■ — 1.
&/,,, "/,)••■/,-••//,i —
Hence, if a £ Z'(it. &\>{m)). we necessarily have
1:(-1)^-^,^.../,..,,=0.
./ -i
and
/+i
/„..,,, =Y(-\)i-lT[h
//11 ^_/ '/ -1/, /„■--;,---/,.
/- i
The last equality is rewritten as
/i l
«('o /,.,) = ^(-1)^^^.../,..,,,/(^,---^---7^
/-1
IH-ll
Let /?„,..„_, = ^/,,/,,-.-/, 1/(^/0 ••• ^/,-1)' and let // be a Cech cochain in
C'-'(U.^P(/»i)) given by {/?/„.../,_,}(/,,<...</,,,)- Then Q = ^ if " G Z'(U.^P(w)).
So /7'(ll.^P(m)) = (0).
(3) Note that C"(U.^P(/»i)) = Z"(ll.^P(m)) = /c[r0 7,,. (7¾ • • ■ T„)-'],„.
Then any Cech «-cocycle a is expressed as f /(To ■ ■ ■ T„)r. where / is a homogeneous
polynomial of degF-ee m + r{n + \) with r > 0. A Cech coboundary (g B"{il.&p{m))
is WF-itten as 5Z"_0(— 1)'./"17"//(7o • •■ 7",,)''. where /,- is a homogeneous polynomial
of degi-ee m + rn and r > 0. If w + r(« + 1) > (r - 1)(« + 1). each monomial
in / is divisible by T- for some i . Hence, we can write f = X]"-()( — l)'77/,- ^
choosing //'s in a suitable way. Thus. //"(P. <fp(m)) = (0) if m > —(« + 1). Suppose
w < — (w + 1). In view of the above observations, we know that //"(P. <fp(m)) is a
/c-vector space isomorphic to
/ / is a homogeneous polynomial of degree
' m + r(n + 1). none of whose monomials
\T{)---Tny is divisible by r., (for each ^. r > o
As its A:-basis. we can choose
I r"1 - - - r„"": Va/ > °" 5Z Q/ = "w f
This implies dim//"(P.<fP(m)) = ( „ ). D
Let us go back to the setting prior to Lemma 5.11. Since /: X <^-» P is a
closed immersion. H'(X.!?) = /7'(P./,^) for each i. i > 0 (problem II.5.3).

120
11. SCHEMES AND ALGEBRAIC VAR1LT1LS
H'iX.SF) is a ^-vector space. We shall show that dimk H'{X.&~) < oo. By
Lemma 5.12. dim,, H'{X.£F) < oo if and only if dim/, //'''(P.^i) < oo. Similarly.
dim*.//''+'(P. 5^-) < oo if and only if dim,,//' + '+l (P..5^, + 1) < oo. Meanwhile.
HH'{P.9r) = (0) (j » 0) by Lemma 5.11. Hence, dim,, H' (X. 9") < oo. By similar
reasoning, we obtain H'{X.^-{N)) = H'{P.i-^{N)) = //'T'(P. 9j{N)) = (0) for
/' > 0 and N ^> 0. These observations are summarized in the following fundamental
theorem.
Theorem 5.13. Let X be a closed subscheme ofP".. and let 3~ be a coherent
&\-Module. Then //'(A\Sr) is a k-vector space of finite dimension for each i > 0.
Furthermore. //''(A\5^(/V)) = (0) for each i > 0 and each N » 0.
Henceforth, we let k be an algebraically closed field, and let X be an -algebraic
variety defined over k. We shall give more details on invertible sheaves on X. A
coherent (f.v-Module 3 is called a locally free sheaf of rank n if there exists an open
covering il = {£/,},€/ such that 3\u, — (<^V,)®" (a direct sum of n copies of @u)
(cf. (1.2.10)). In particular, if n = 1. we call 3 an invertible sheaf on X. \i 3
and Jt are invertible sheaves, then 3 ®e,x Jt is also an invertible sheaf. If we put
S?~l := SP*mrx {3.&x)- %~X is also an invertible sheaf. It is clear that if 3 = 3'
and Jt = Jt1. then 3 ®ffx Jt = 3' ®„x Jt' and 3"x = 3'~]. (Compare the
definition of <f*-(/) in this chapter.) Denote by [3] the isomorphism class of 3.
Then the set of these isomorphism classes becomes an abelian group with respect
to the multiplication [3] ■ [Jt] = [3 <g>ffx Jt]. the inverse [3]~l = [3~l] and the
identity element \@x\ Denote this group by Pic(A') and call it the Picard group of
X.
Let 3 be an invertible sheaf on X. and let il = {£/,},e/ be an open covering of
X such that 3\Vi = <fUi (i G I). We say that the open covering il trivializes 3. Let
<Pi: 3\u, —> &u, be an isomorphism. Since 3\v, is a free (fj,',-Module, we may write
2C\l\ = $u,<-'i with a free basis e,-. Then </>,- is a mapping aet i—> a. The isomorphism
<Pi • ip~x: (f(';n(.', —> ^v,nur which is obtained by restricting </>,. </>,- onto £/, n £/,-. is
determined by the image 0,-,- of 1 G r(£/, n £//.(¾). where 0,-,- G r(£/,- n Ur@x)* =
the multiplicative group consisting of all invertible elements of r(£/, n Uj.tfx). We
have the equality 0,,, = 0,,,- • 0,-,- on £/,- n £/,- n £/,,. where 0,-,-|t',nc,nt4 is abbreviated
by 0,-,-. Moreover, we have 0,, = 1 for each /'. If we alter the isomorphism ip/ to
i//j: 3\i\ -^ ^L; (i G I), the isomorphism t//, • <pfl: <fL.. -^» ^- is determined by the
image rr,- G T(£/,-.^)* of 1 G T( £/,-.(¾). Hence, if y/j • y/,"1: ^L-nc, ^ ^c,nc, gives
rise to /;,, G r(f/, n £/,-.<$>)*. P\\ is related to 0,-,- as
Pit = (T/k',nc,)"' -0// • (tf/|c,nt/,)
which we abbreviate as /;,, = f, ' • 0// ■ "V-
Replace the open covering il by a refinement 9J = {F;}/eA. Choose a mapping
a: A —> / so that I7; c £/„(;.) for each A G A. If / = a{X). the isomorphism
</?,-: jzf|[.- -^+ (ft-, induces an isomorphism </?;.,-: ^f|,^ -½ ^,-. If / = c*C"). W/ •
V/7/' : ^i',nr„ —> <^V,nr„ is given by the restriction 0//|r,nr„-
As explained above, an open covering il = {£/,-},-e/ and the choice of
isomorphisms (/3,: ^1(-, -^> &i\ (i G /) give rise to a family of functions {0/,-},-./e/ which
we call -a family of transition functions. The family {0,,},_,e/ defines an element 0U
of the cohomology group //'(il -^)- where @% is the sheaf of multiplicative groups
associated with U i-> r(£/. ^j)*. A different choice of isomorphisms {y/,-},-e/ alters

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES 121
the family of transition functions {0/,}/./e/ only by a coboundary element as shown
above. If it ranges over all open coverings of X which trivialize 5?. then the elements
Oix define an element 0 in the Cech cohomology group H](X.<fx) = lim //' (U. (¾).
Conversely, an element 0 of HX{X.@X) K given in terms of an open covering
it of X and an element On e H[(ii.&x)- Represent On by a cocycle {#,,}, /e/ with
0/i G r(t/,- n Uj.@x)*. Patch &i\ (i G /) together to define an ^-Module Se by
the following identification:
/,■ ~ /,• if and only if /,■ = {6,i\v) ■ /, for U C U,■ n £/,. /,- G
r(t/.<f,,.,). and/, G r(t/.(ft',).
Then 3? is an invertible sheaf. Different choices of it and different ways of
representing 0 by cocycles {0/,} give rise to the same invertible sheaf up to isomorphisms.
We have, therefore, proved assertion (l) of the following lemma.
Lemma 5.14. (1)-4 mapping3 i—> {0,/} gives rise to a one-to-one correspondence
between the ahelian groups Pic(A') and H](X.tfx).
(2) Let SC and J? he invertible sheaves on X. let it = {£/,},-€/ be an open covering
of X which trivializes 3 and Jl. and let {0,,} and {r,,} be families of transition
functions of 5? undJl. respectively. Then {0,, • r,-,} and {0,71} arc families of transition
functions of 5C &W\ J? and 5? ~'. respectively. The correspondence in (1) is therefore
an isomorphism of groups.
A proof of the assertion (2) is straightforward. In order to define a family of
transition functions {0/,} of 5f. we made use of the isomorphisms </>,■: 3,\i;l —> &v, ■
where </>, is a mapping ae, i—> a if 5f\ci = @c,l>j with a free basis r,. Since 3f\v,, =
&i\e-, = ^inCj (£/,, = Uj n £/,). we have e, = ajjCj with a,, G Y(Uij.@x)* and
aki = akj -otji. {a,,} is related to {0/,} determined by {</>,} as a,, = 0,, = Of,1.
We say that {a,,} is also a family of transition functions. The readers should be
careful not to confuse the transition functions by free bases (i.e.. {a,,}) with those
by coefficients (coordinates) (i.e.. {0/,-}).
Next we assume that A' is a normal algebraic variety. We consider all (irreducible
and reduced) closed subvarieties of codimension 1 of X and the free abelian group
Div(A') generated by them as free basis. An element of Div(A') is written as a finite
sum '^2,lrijYj. where «, G Z and Y, is a subvariety of codimension 1. We call an
element of Div(A') a Weil divisor (or simply a divisor). If n,- > 0 for each i. we call
D = ^2; n, Yj an effective divisor. D > 0 by notation. If D = Y, for some i. we call
D an irreducible divisor. When D > 0. we call D a reduced divisor if «, = 1 for each
i. Given a nonzero element / G k(X) we define a divisor (/) associated with / as
follows.
Let Fbea subvariety of codimension 1 of X. and let n be the generic point
of Y. We denote the local ring &x.i) by &XY- Since X is normal by hypothesis.
&x y is a discrete valuation ring of k(X). Let »y be the associated discrete valuation
of k(X). vy is uniquely determined if we put vY(t) = 1 for a generator t of the
maximal ideal mx.Y of^x.Y- We then put (/) =Y.y v >'(./) ^- where Y ranges over
all subvarieties of codimension 1 of A\ We have to verify that ^2Y vy{f) Y is a finite
sum as follows. Let U = Spec(y4) be an affine open set of X. Since X — U is a closed
subset of codimension > 1. there are only finitely many irreducible components of
codimension 1 in X - (J. On the other hand, since k(X) = Q(A). f = h/a with

11. SCHLMLS AND ALGEBRAIC VAR1HT1LS
a.b G A. Replacing A by ^[¢/-1]. we may assume that A is a finitely generated k-
algebra and f & A. Let Y be a subvariety of codimension 1 of X. let rj be its generic
point, and let & = &x.,, = ®x.y ■ If r] £ U. then Y is an irreducible component of
X - U. Suppose y] G U. Then & = ^p(ht(p) = 1). If f G (Av)*. then vY(f) = 0.
and if / G pAp. then vY(f) > 0 and p is a prime divisor of f A. Conversely, if p
is a prime divisor of f A. then htp = 1 and Ap is a discrete valuation ring (1,1,33),
Let Y = {p}. Then vY(f) > 0. By this observation, we know that J2y vy(/)Y 's
a finite sum. We call Y a zero divisor of / if vy{f) > 0 and a polar divisor of /
if vr(f) < 0- The absolute value |v;-(./)1 if called the index of a zero divisor or a
polar divisor. Set (/)0 = £,., (/)>()i-y(/) Y and (/),^ = - E,, (/)<()«>-(/) F- We
call (/')() and (./)30 the zero part and the polar part of/, respectively. It is clear that
(/)() and (/)oo are effective divisors and (/) = (/)0 - (/)00. If we emphasize that
(/) is considered on X. we denote it by (/)jf. By the definition, it is apparent that
(/#) ~ (/) + G?) and (/-') - -(/) for /.g e k{X)*. We therefore obtain a group
homomorphism <p: k{X)* —> Div(A'). / h-> (/). By definition, two divisors D and
D' are /mtW.v equivalent, written £> ~ £>'. if £>' - Z> = (/) for / G Ar(A')*. (Verify
that £>' ~ Z) is an equivalence relation.) We put CI(X) = Di\{X)/<p{k{X)*) and
call C/(Ar) the divisor class group X. We have the following result.
Lkmma 5.15. Let X be a normal algebraic variety defined over k. Then Kery? =
T{X.@x)* f01' the above homomorphism <p: k(X)* —> Div(A'). In particular if X is
complete, then Ker</> = k*.
Proof. Suppose (/) = 0 for / G k(X)*. Let x be a point of X. and let
£/ = Spec{A) be an affine open neighborhood of x. Then A = p)^p- where p
ranges over all prime ideals of height 1 of A (cf. (1.1.33)). For such a prime ideal
p. let Y = {p}. Since vY{f) = 0 by hypothesis. / G <f£ y = A*p C Ap. So / G A.
Similarly. /-1 g ,4. Hence. / G /1*. Since .v is an arbitrary point of X. we know
that / e r(A\<fV)*. It is clear that T{X.<fx)* Q Kery. If X is complete, we have
Y{X.cfx) = A: by Theorem 4.18. Hence, the last assertion follows. □
Suppose A' is a normal algebraic variety. Given an invertible sheaf J? on I.
take an open covering it = {U-,},€;. isomorphisms ip:: Stic, —> @v, ■ and a family of
transition functions as above. Fix an index /0 G / and denote it simply by 0. Put
(hi0 =^(0 = //- Then we can regard/,-.0,-,- as elements of k {X) satisfying the relation
f I = Ojifi- The function /,- may have zeros or poles on £/,- — £/,- n £/<). Define a divisor
D, on U, by D, = (./)(-,- Since 0r, G T(£/, n Urcfx)*. we have D,- = D, on £/,- n £/,.
Hence, there exists a divisor D on X such that D, = £)/, for each i G /. Replace
the index i0 by i'i and set g, = f-,\ and h = /,-„;,. We have g, = /,-/1 as-elements
of k{X). Let £>' be the divisor given by {#,-},-<=/. Then £)' = £) + (/1). So ZV ~
D (linear equivalence). Alter the isomorphisms <fj\ St\it ^> tf\ji to isomorphisms
1//,: St\yl -^+ cfL'r According to the previous notations. />/7 = ctJ[ ■ By, • a-,. Fixing
an index /0. put anew g, = /;,,-,. Then as rational functions on £/,. g, = 0/1/,0¾
with (To = ^/,, G A;(A'). Since rr,- G r(£/,.^)*. we have (#,-) = (/,-) + (a,,) on £/,-. So
D' = D + (rr0) if Z)' denotes the divisor given by {g,-},-e/. That is to say. D' ~ Z). If
we replace the open covering it by its refinement 2J. the divisors given in the above
way by means of 2J from St are linearly equivalent to D. Thus St defines the linear
equivalence class [Z>] of Weil divisors. We call its representative D "the" Weil divisor
defined by St. We denote D also by D{St).

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VAR1ET11 S 123
It is easily ascertained that the mapping w: Pic(l') —> CI(X). [L] h-> [£>]. is a
group homomorphism. Suppose [D] = 0. Then D = (h) with h G /c(A')*. According
to the above notations. /)h~[ G r{Uj.$x)* for each i G /. Set .v, = f, • h~[. Then
Qji = .v, • .s/'. This implies 5C — &x- Hence, the homomorphism w is injective. In
general. \m(w) C Cl(X). In this respect, we call a divisor D a Cartier divisor if D
is linearly equivalent to the divisor D{5C) which is defined by an invertible sheaf 3?.
Let R be a normal ring. We can consider the free abelian group generated by
all irreducible and reduced closed subschemes V{p) of codimension 1 of Spec(/?).
where p G Spec(R) and ht(p) = 1. To / G Q{R)* we associate (/) = £p up(/) F(p).
where p ranges over all prime ideals of height 1 of R and vp is the discrete valuation
of Q(R) associated with a discrete valuation ring Rp. We can show as follows that
Sp vr>(f) ^(p) is a finite sum. Suppose firstly that / G R. Then vp(/) > 0 for each
p. and Vp(f) > 0 if and only if p is a prime divisor of JR. Hence. (/) is a finite
sum provided / G R. If / ¢ /?. write / = a/b (a.b G /?) and (/) = (a) - (/>).
Thus, (/) is a finite sum in view of the first case. In the same way as we define the
group Cl(X). we can define the linear equivalence of divisors and the abelian group
C/(Spec(/?)) of all equivalence classes of divisors. We abbreviate C7(Spec(/?)) as
Cl(R). The ring R is a unique factorization domain (UFD. for short) if and only
if Cl(R) = (0) (problem II.5.5). that is to say. every prime ideal of height 1 is a
principal ideal.
Now assume that every local ring &x.x of an algebraic variety I' is a UFD. There
are plenty of algebraic varieties satisfying this condition. If we employ a definition
of the next chapter, we can refer to the following result of Auslander.
Lemma 5.16. A regular local ring (R.m) (cf. Definition 6.10) is a unique
factorization domain.
Every local ring of a nonsingular algebraic variety defined in the next chapter is a
regular local ring: hence, a UFD. Every local ring of X is by hypothesis a UFD. So it
is a normal ring. X is therefore a normal algebraic variety. Let D be a Weil divisor on
X and write D = n\ Y\ + ■ ■ ■ + n, Y,.( Y,: irreducible. «, G Z). Let x be a point of X.
Since the local ring &x.\ is a UFD. each Y, is defined at x by a principal ideal ai@x.\-
where a, G &x v if x ¢ Y,. Put /\ = FJ'= i a",' which is in general a rational function.
We can consider that /v defines in an open neighborhood t/ of x the divisor D
with the multiplicity «, of each irreducible component Y, counted in. {£/ : x G X}
is an open covering of X. We denote by U = {£/,-},-e/ this open covering and by /,
the defining equation of D corresponding to £/,. Then /, • //' G r(£/, n Ur@x)*-
Let Oji = fj • .//1. and let Sf be the invertible sheaf defined in terms of il and
{Oji}. It suffices, for example, to set S?\c, = &u, • .//1. We call this Sf the invertible
sheaf defined by D and denote it by @x{D). If we choose another family of local
defining equations instead of {/,■}, we obtain an invertible sheaf which is isomorphic
to tfx(D). We have thus proved the following lemma.
Lemma 5.17. Let X be an algebraic variety whose local rings are all unique
factorization domains. Then the mapping 5? ^ D{5?) is an isomorphism between the
abelian groups ?\c{X) and Cl(X).
Example 5.17.1. For P = P'/ we have Pic(P) = Z[<fP(l)].
It suffices to show that @p(D) = @v{d) (d G Z+) for every irreducible closed
subvariety D of codimension 1 on P. D is defined by a homogeneous prime ideal *}3 of

124
11. SCHEMES AND ALGEBRAIC" VARIETIES
height 1 of the homogeneous coordinate ring k[T(} T„] of P. Since k[T() T„]
is a UFD. there exists a homogeneous polynomial F such that ^- (F). Consider
the canonical open covering P = U/'-o ^ w'tn ^< = DAT,). D is denned by
F{T0/Ti T„/T,) = F ■ Tr'i (d = degF) on £/,, Hence. 0p{D)\l; = ffv, • (Tf/F).
So the transition function on £/,-, is {Tj/TiY. It follows that <fP{D) = <fP{d).
Let I be a normal projective algebraic variety, let jS? be an invertible sheaf on
X. and let D be the divisor defined by 5C. Let a be a nonzero element of H°(X.5C).
Take an open covering it = {£/,};<=/. isomorphisms </>,-: 5C\vj ^> <?v, and a family
of transition functions {0,-,-} as specified as above. Let .v,- = Lpj{a\Ui). Then .v, G
r( ¢/,-.(¾). and .v,- = OjiSj on £/,- n £/,-. Hence. .v,-/r' = .v,-/,"1. where /,- = 0,-,-,
according to the previous notations. Denote this element by h. Then h G k(X)
and Sj = /', • h for each / G I. We have an effective divisor (.y,-) on £/,- and patch
{(.v,-)},-6/ together to give an effective divisor £)' > 0 on A', where D'\u, = (.">'/) ar)d
D1 = D + (A). We write Z>' = Z>(rr). The divisor D(a) is determined only by 5?
and <7 independently of the choice of D. We denote D(a) also by (c)o-
Suppose D(it) = Z)(t) forrr. t G H°{X.5C). For r,- = <^,(t|6- ). we have/, = 0,,/,.
Since D(fr)|t,. = £>(t)|l- . /,- -.vf1 G r( ¢/,-.(¾^ for each i G / (Lemma 5.15). t, -.v~'
is an element of T(X.&x)*{= k*) which is independent of i. Namely. /,- = as,- with
a G k*. Hence, x = an.
Conversely, let D' be an effective divisor on X such that D' = D + (h) with
h G k(X). Since /,-/i = 0 is a denning equation of D'|r, ■ we have f)h G r(¢/,-.(¾-)
for each / G /. On £/, n £/,. we have
<Pi ■ ¥»r'(//A) - 0// • //A = //A = <p,- ■ <p;\fjh).
Hence. <pf[(f,h) = <fif[(f' jh) (for each i.j G I). So there exists a G r(lr.^-) such
that <j>i(a\u) = /,-¾. Through these observations, we have proved the next result.
Lemma 5.18. Let X be a normal, projective algebraic variety, let 5? he an invertible
sheaf on X. and let D be the Cartier divisor defined by 5C. Under the mapping a i—>
D(ct). the set H(){X.5C) — (0)/(~) is then in one-to-one correspondence with the set
{D':D' >0.D' = D + (h) for some h G k(X)}. where a ~ x if and only ifx = aa
[for some a G k*) for a. x G H°{X. 5?) - (0).
H[){X.£?) is a finite-dimensional A:-vector space. Choosing a basis {(T0 a„)
(« + 1 = dimkH"{X.&)). we have H^X.SC) = k"+K Thus, the set H°\x.5C) -
(0)/(~) is identified with the projective space P" of dimension n. Let D = D(5C) and
let |D| := {D':D' > 0. D' = D + (h) for some h G k(X)}. We call \D\ the complete
linear system of the divisor D. \D\is identified with P" by the above lemma. We call
H°{X.£?) the k-module associated with \D\. For a vector subspace M of H°(X.J?).
define a subset L of |£>|. which we also denote by L(M). by L = {£)' G \D\: D' = D{a).
a e M - (0)}. As a set. L is isomorphic to M - (0)/(~) which is a linear subspace
of the projective space P". Conversely, take a linear subspace L' of P" = \D\. Then
there exists a vector subspace M' of H°{X.5?) such that L' = L(M'). Since M' is
uniquely determined by L'. we denote M' also by M{L'). We call L a linear subsystem
(or simply a linear system) of \D\ and M = M(L) the associatedk-module of L. The
set {h G k{X): [h) = D{a) - D. n G M - (0)} U (0) is a subset of k(X). which is
a k-vector space isomorphic to M. We call this space also the associated k-module
of L. We put dim/, L = dimA M — 1 and call it the dimension of L.

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES 125
Suppose H(){X.£?) £ (0). Write ST = @x{D). We shall consider a linear
subsystem L of \D\ and its associated ^-module M. We assume dim/, M = r + 1 > 0
which is equivalent to the condition L ^ 0. We regard M as contained in k(X) and
take its A;-basis {/o /,•}. where each /, is a rational function on X. We then
define a rational mapping 0),: X » P' by P h-> (/0(/5) //-(^)) G P' where P
is a closed point of A'. We can ascertain as follows that ¢/ is a rational mapping. Let
U = Spec(A) bean affineopen set such that for each i. /, &A = r(U.$x)- Moreover.
we may assume that /0 / 0 everywhere on U. If 7~o T, are homogeneous
coordinates of P'. the restriction of ¢/, onto U is the morphism of schemes U —>
A' = D\-{To) <^-> P' attached to a ring homomorphism k[T\/T{) Tr/T()] —> A.
Tj/Tq h^ /,//()• Thus. ¢^ is a rational mapping. We call ¢/ the rational mapping
associated with L.
Take a different k-basis {go £r}- Then #,■ = X)/--oa'///' (0 < ' < '')• where
(a,,) G GL(r + 1. A:). If we use {go #,.} instead of {/0 /,■}. the associated
rational mapping is /„ • ¢/. where /„ is the projective transformation of P' defined
by the matrix a = (a,/).
The A;-basis {/0 ./, } is a A;-basis of M which is identified with the k-module
MD : = {h e k(X): [h) ='D{a) -D. a G M - (0)} U (0). If D1 is a divisor linearly
equivalent to D. then |D| = \D'\. If we write D' = D + (u) with u e k{X). then the
£>module Mly := {^ e k{X); (g) = D{ct) - D'. a G M - (0)} U (0) is isomorphic
to Mp via g ^^ gu. Hence, if {ga g,} is a A;-basis of Mo'■ then {gttu g,u}
is a A;-basis of MD. The associated rational mappings P t-> (jjo(-P) gr{P)) and
P^ (^(^)(^(/1) ^,.(^)^(/1)) are the same.
The above observations show that the rational mapping ¢/ : X ■ ■ ■ —> P' is
uniquely determined by the linear system L up to projective transformations of P'.
For two divisors D\.Di. we write D\geDi if* Z>i — Di > 0. For a linear system
L(C \D\) and an irreducible divisor A. we say that A is a fixed component of L
if Z)' > /1 for each D' e L. If L 7^ 0. it is apparent that we can determine an
effective divisor F in a unique way so that D' > F for each D' G L and F is the
largest (with respect to >) among those effective divisors with this property. We call
F the fixed part of L. Then L — F := {D' — F:D' G L} is a linear system again.
and dim(L - F) = dimL. (In fact, since |D| = |Z)0| for an element D0 of L. by
replacing D by Do. we may assume from the beginning that D > 0 and D G L. Let
M{L) := {/ G k(X):D + (/) G L} U (0) be the associated A;-module of L. Then
D'-F = {D-F) + {f) for D' e L. Hence. L - F C |£> - F | and M (L - F) = M (L).)
We call L — F the variable part of L. Even if L has no fixed components, i.e.. if F = 0.
there may still exist a point P G D' for each D' e L. (In general, when we write
an effective divisor D as a sum of irreducible components D = ]T\ «, F, with each
«, > 0. we denote 1J( Y, by SuppZ) and call it the support of D. We write P e D'
to mean P G SuppZ)'.) We call P a tow /w/«/ of L it' P £ D' for each Z>' G L. Set
BsL = {P G X: P is a base point of L}. Then BsL = Hce/ SuppZ)'. This relation
holds in the case F > 0 as well if we regard the points of fixed components as the
base points of L. BsL is a closed set of X. and codim^ BsL > 2 provided F = 0.
Suppose a rational function / satisfies the condition D + (/) > 0 for an effective
divisor D = J2i n< Yi with each «, > 0. If Y is a polar divisor of /. then Y = Y,
and vy(f) > —H,- for some /. So the index of Y as a polar divisor of / is at most
«,. We shall prove the following lemma.
Lemma 5.19. Let X be a normal, projective algebraic variety. Then we have:

126
11. SCHEMES AND ALGEBRAIC VARIETIES
(1) Let D be a Carrier divisor, let LC\D\ be a linear system with dim L > 0. and let
<£>l be the rational mapping associated with L. Then <bL is defined on X — Bs(L — F).
where F is the fixed part of L.
(2) Let p: X ■ ■ ■ —> Y be a dominant rational mapping from X to a projective
algebraic variety Y. Then there exists a linear system L such that p = <t>£.
Proof. (1) As explained above. M{L) = M(L- F). and ¢/. = Q>l-f- We may
as well assume A G L and L C |£>0|. Accordingly, we may assume that 1 G M(L)
and {l./i //}('' = dimL) is a A;-basis of M(L). If P ¢ Supp A. we claim
that the rational functions f\ /,• are regular at P. Set f = /,■ and & = tfx.i>-
Suppose / g cf. Since & = Hp^p wrtn P ranging over all prime ideals of height 1
of &. there exists some p G Spec(<f) such that ht(p) = 1 and f ¢ cfp. Since <fp is
a discrete valuation ring, then /-1 G p<fp. Let 7 be the irreducible divisor {p} on
X. Then vy(f) < 0 and P is in a polar divisor Y of/. However, the hypothesis
A) + (/) > 0 implies that every polar divisor of / is an irreducible component of
A)- This contradicts the assumption P 0 Supp A- Hence, for each i. /, G <^V./>.
This means that ¢/. is defined at the point P. Suppose P ^BsL. Then P ¢ Supp A
for some i. where A = A + (//) (1 < ' < 0- (Since £> is a Cartier divisor by
hypothesis. D is defined at every point by a single equation. This is the case for a
divisor Z)o linearly equivalent to D. Let g = 0 be a defining equation of A is an open
neighborhood of the point P. Then A is defined by gf, = 0. and P G Supp A If and
only if (gfi){P) ~ 0. Suppose P G Supp A for each /. Then {gf)(P) = 0 for any
element / of M{L) because / is expressed as f — ao + ot\f\ + ■ —h <*;-/,• (a, G fc).
Namely, if P £ Bs L. then P £ Supp A for some /.] Then {1///. /i///.' • • • ■ /,•///}
is a /c-basis of the fc-module {g G £(^): (g) = D - A- £> 6 £}• By the above
prescription, we know that ¢/ is defined at the point P. So. X - Bs(L - F) is
contained in the domain of definition of <!>£..
(2) Y is a closed, algebraic subvariety of a projective space, say P". Replacing
P" by a linear subspace if necessary, we may assume that Y is not contained in any
hyperplane. Let <fy{\) = cf\>(l) ®&y From the description given after Corollary
5.10. it follows that the homomorphism i : H'°'(P.&p(l)) — H0(Y.cfY{l)) induced
by the closed immersion r, Y —> P" is injective. On the other hand, let U be the
domain of definition of/?, and let ip = /?|l<. J? = <p*cfY(\) is then an invertible
sheaf on U. Let j: H°{Y.cfY{\)) —> H^iU.S1) be the natural homomorphism.
(Compare the definition of the inverse image in Part I, Chapter 2.) M = Im(/ • i)
is a finite-dimensional /c-module of H[)(U.5C). A nonzero element a of M defines
an effective divisor D'(a) on U. If we write D'(a) = ^2/niZ', then define D{a) by
D(a) = Y^i niZ-. where Z,- is the closure of Zj in X. We also write D(a) = D'(cr).
Then L = {D{a);cr G M — (0)} is a linear system on X and the associated k-
module of L is equal to M. In fact, fix D'{) = D(a()). Then D'{a) = DQ + {/„),
f„ G k(U) = k{X). Since codim^(Z - t/) > 2. we have D{a) = A -+ '(/).
where A = £>("o)- So L is a linear subsystem of |A| and M(L) := {/^: <r G
M — (0)} U (0) is /c-isomorphic to M. Let 7¾ ^,, be homogeneous coordinates
of P". and let <t0 "■« be the images of To T„ by j • i. Choose /, (1 < / < «)
in such a way that /)((7,) = D(fr0) + (/,). Then /? is the rational mapping given by
P => (l./i(P) /„(P)). If we prove that l./i /„ are linearly independent.
we conclude /; = ¢/,. Suppose a0 + "i/i + • ■ ■ + a„/„ = 0 (a/ G fc). Then /;(f/)
(accordingly. Y = p(U)) is contained in the hyperplane aoPo+ ^1^1 +• • ■ + a„T„ = 0.
By the initial hypothesis that Y is not contained in any hyperplane of P". we must

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES 127
have q() = ■ • • = a„ = 0. Namely. 1. f \ /„ are linearly independent. □
With the notations in the above proof (2). M = //°(P. <fp(l)). An element a
of M — (0). viewed as an element of //°(P. ^p(l)). defines a hyperplane H(o). We
then say that D(a) is the pullback of H(a) by /;. We write D{a) = X • H(o) to
simplify the notation.
Corollary 5.20. With the notations of Lemma 5.19 (1). Bs(L - F) = X - U{).
where Uq is the domain of definition of <t>/.
Proof. We have shown Bs(L - F) D X - U{). Let p = <S>L: X -> P" (n = dimL)
and let Y = p(Uo). p: X ■ ■ ■ —> Y is then a dominant rational mapping. Furthermore.
L - F = {X ■ H{(t)\(t e M - (0)} by Lemma 5.19 (2). /;|((1: £/„ -^ /;(£/„) is a
morphism of schemes, and the linear system of hyperplanes on P". {H(a):a e M —
(0)}. has no base points on p{Uu). Hence, any common point of {SuppD(a);a 6
M — (0)} lies outside £/0 even if such a point exists at all. Namely. Bs(L — F) C
X - £/„. □
Definition 5.21. Let 3? be an invertible sheaf on a normal, projective
algebraic variety X. When H{){X.5C) ^ (0). we denote by ¢^ the rational
mapping <$>A: X ■ ■ ■ -^ PN (N = dim A) defined by the complete linear system A : =
{D{a):cr 6 H(\X.Se) - (0)}. We say that S? is a very ample invertible sheaf if
Bs A = 0. and ¢^ is a closed immersion. We say that jzf is an ample invertible sheaf
if Jz?x"'(= 5C ¢0 ■ • ■ CO 21) is very ample for some positive integer r.
Example 5.21.1. Let X be a closed algebraic subvariety of P" which is normal,
and let jzf = &v{\) <a@x. Then 3? is a very ample invertible sheaf.
With the notations of Lemma 5.19. set X = Y and/; = id*. M is a /c-submodule
of H(\X. 5C). Replacing P" by a linear subspace if necessary, we may assume that X
is not contained in any hyperplane. Then M = H°{P.@y>{\)) though M g H[){X.5C)
in general. So if L is the linear system determined by M. O/ = n • <$>2 . where n
is the projection from the projective space P^ (A^ = AimH{){X. 5f) — 1 > n) to
P". {T0 TN) ^ (T[) T„). Since BsL = 0. the complete linear system A
that contains L has no base points. Hence. d>j2 is a morphism. If d>/ is a closed
immersion, then ¢^ is a closed immersion, too (problem II.5.6).
As a final topic of this chapter, we shall discuss the normalization of a projective
algebraic variety. Let A be a graded ring such that A$ is a field k (not necessarily
an algebraically closed field) and A is a finitely generated /c-algebra domain, and let
L be its quotient field.
Lemma 5.22. Let A he as above. Then there exists a positive integer N such that
Proj(^) is isomorphic to Proj(B). where B is a graded subalgebra of A that is generated
by An over Aq.
Proof. Noting that A is finitely generated over k. take generators {.\"i v,,} of
A which are homogeneous elements. Let d, = deg.v,. For each i. A[x-~ ]o is finitely
generated over k (Lemma 5.5 and the proof of Theorem 5.4 (3)). and we can take
generators of the form Y[ /, -v?'/-v/ (ai >: 0. /i > 0. /?</,■ = Yl,/i a/dj). With i fixed,
let t'i be the least common multiple of the/? d, when all generators are counted in. Let
N be the least common multiple of d\ d„. e\ e„. Let B := k[AN]. Then B is a
graded subalgebra of A. and the natural injection f.B—>A gives rise to a morphism

128
II. SC'HF.MES AND ALGFBRAK" VARIETIES
of schemes f: G(ip) —> Proj(6). Since a, ' G AN for each i. we know that G{ip) =
ProjU). Furthermore. Proj(S) = \J! ^^x^''1). £>., (a,) = /-' {D . (.vfM)) and
Proj(^) = (J/'-i ^+ (-v')- Hence, in order to show that / is an isomorphism, it suffices
to prove that the homomorphism ip:: 5[(A-f'''')_1]o —> A[x^1]q induced by ip is a k-
isomorphism. Since <p/ is clearly an injection. ip-, is an isomorphism if the generators
of ^[.\-r']() as a k-algebra are contained in B[(xi )~']o. As remarked already, such
a generator is of the form EI//, x'j' /xi ■ Since fid, J N. we have
(/3, is. therefore, an isomorphism. □
Lf.mma 5.23. Let A be as above. If A is integrally closed, then Pro}(A) is a normal
scheme.
Proof. As in the proof of Lemma 5.22. let A = k[x] v,,]. In order to show
that Pro)(A) is a normal scheme, it suffices to prove that y4[.\-r']() is an integrally
closed, integral domain for each i. Let £ be an element of the quotient field K =
Q{A[.\,"']()). We can write £ = u/v (u. v G A„ (for some n > 0). v ^ 0). Suppose £,
is integral over y4L\-r']0. Then there exist an integer r > 0 and a monic relation
{xtf)N + </i(.v;-£)'v-' + ■--+^=0. a,eA.
Since A is integrally closed, w : = *,-'<; G A. From the equality vw = .yJ'u. we see that
w is a homogeneous element of A of degree r • degx,. So £ = w/x- G ^[.Y^'jo. □
We assume below that A is generated by A\ over k. An element £ of the quotient
field L of A is said to be a homogeneous element of degree p — q if <!; is written as
s = f /g. f e Ap. g <E Aq. g ^ 0. Let L, = {£ G L: £ is a homogeneous element of
degree /}. Then the relation L, • Ls C L,+v holds. Hence. L := ^,.6Z £, is a graded
ring which has homogeneous elements of negative degree even if A does not.
Lkmma 5.24. With the above assumptions and notations, we have
(1) Let x be a nonzero element of A \. Then x is algebraically independent over
L{). L = L()[a'..v~']. and L = Lq(.x).
(2) Let A be the integral closure of A in L. Then A is a graded ring such that
A-, = 0 for each i < 0. Moreover. Ay is a finite algebraic extension of k and A is
finitely generated over k.
Proof. (1) First of all. note that La is a field. Suppose x is algebraic over L().
Then there exists a relation
iH).x'" + a\.x'"~[ + ■ • • + a„, = 0. a, G Ar for some ;• > 0.
Since deg«,-.v"'-' = m — i + r. every monomial aiX'"~' must be 0. Since A is an
integral domain, each a-, = 0. This contradicts the hypothesis that a be algebraic
over Lo. x is. therefore, algebraically independent over L0. Let / G Ap. Then
f/.x1' e L(). Hence. /' G L()[.v]. Suppose f/g G L, (/' G Ap.g G At/). Then
f/x = {fhp) ■ (x"/g) • x''~" G LoI-v.a-1]. So L C LoIa-.a--']. The opposite
inclusion is clear. If we note that A C L. we know L = Lo(a-).

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES
129
(2) By (1) above. Lo[x] is a polynomial ring in one variable over the field La. In
particular. Lo[x] and Lq[x. .v_1] are normal rings. Since A C L{)[x]. we have A C L0[x].
Write an arbitrary element d; G A as
£ = £v + C,,! + • ■ ■ + 6- C, e Lo-y'. .v > 0.
If each c,t 6 ,4. we know that A is a graded ring. Since £, is integral over A. A[q\ is
a finitely generated /4-module (1.1.10). Hence, there exists z g /4(/ (r ~£ 0) such that
r/4[d;] C A. So r • £,' g A for each i > 0. Write - • c' = r • c,{ + ■ ■ ■. Comparing
this with the decomposition of z • £,' into homogeneous elements in A. we deduce
that z ■ £,[ e A. Namely. A[c,x] C A • (1/r). Since A is a noetherian ring. A[£,] is a
finitely generated ,4-module. This implies £, G /4. Then £ - c, = £4, i + • ■ • + Q, G A.
Repeating the above argument, we know that each £/ e A. Meanwhile. A is a finitely
generated ^-algebra by (1.1.34). If we note that /J<) is integral over k. (1.1.10) implies
that /4o is a finite algebraic extension of k. From /4 C La[x]. we conclude that /4, = 0
for each /' < 0. □
Combining the above results, we obtain the following result.
Theorem 5.25. Let X be a projective algebraic variety over a field k. Then its
normalization X (cf. (4.23)) is a projective algebraic variety, too.
Prooe. We can write X = Proj(/4) for a graded ring A over a field k which is
a finitely generated ^-algebra domain. We may assume A() = k and A = k[A\]. Let
A be the integral closure of A in Q{A). By Lemma 5.24. A is a graded ring which
is a finitely generated ^-algebra domain. A0 Q LQ = k(X). and A0 is an algebraic
extension of k. Since k(X) is a regular extension of k. we have A() = k. A is.
however, not necessarily generated by A\ over k. But. by Lemma 5.22. there exists a
graded subring B of A such that B is generated by B^ over k. which itself is a finitely
generated ^-module, and Proj(5) = Proj(/4). By Lemma 5.23. Proj(5) is a normal
scheme. If we can show that X = Proj(5). then we know that I is a projective
algebraic variety over k. (Since B = k[Bx]. we have only to alter the grading of B
so that the elements of BN have degree 1.) We show that X = Proj(5). The function
field of Proj(5) is equal to the quotient field of Proj(/4). and the function field of
Proj(/4) is contained in Z,() = k{X) because A C Lq[x]. The opposite inclusion also
holds. So we have A:(Proj(£))^ = k{X). Write /4, = kx\ + -■ + kx„. Let ^ be a
homogeneous prime ideal of A such that each a; e ^3. Then ^3 D A^. Since A t is a
maximal ideal of A. we have A+ = tynA. Hence. ^ is a maximal ideal of A as well.
So $ = /4 + . Moreover, we have Proj(/4) = U"_| £>+ (.v,). If we show that /4[a;~']0 is
integral over A[xj~l]0. it follows that Proj(/4) —> Proj(/4) is a finite morphism. But.
/4[.v,~']o is integral over A[x~']() because A is integral over A. Then Theorem 4.23
implies X ^ Proj(i) ^ Proj(fi). D
II.5. Problems
1. For a graded ring A = V]„>0/4„ and an ideal I of /4. show the following
assertions:
(1) The condition that I is a homogeneous ideal is equivalent to the condition
that for each .v e I. if .v = a-() + ai + • • • with a, g A,-, then a, g / n/4, := /, for
each /.
(2) If / is a homogeneous ideal, then so is the radical \/7.

130
11. SC'Ht.MhS AND ALGLHRAIC VARU.Tll.S
2. Suppose a graded ring A = '}ZI,>0A„ is a noetherian ring. Applying the
arguments in Part I. Chapter 1 and following tht_ subsequent steps, show that each
homogeneous ideal has a homogeneous primary ideal decomposition.
(1) A proper, homogeneous ideal I of A is written as the intersection of finitely
many, irreducible homogeneous ideals I = I\ n ■ • • n/„.
(2) An irreducible homogeneous ideal is a primary, homogeneous ideal.
(3) An arbitrary homogeneous ideal 1(^= A) has a homogeneous primary ideal
decomposition I = 0\ n • ■ • n £3,,. as in (1.1.23). satisfying the following two
conditions:
(i) v/37^ v/57 if/7^./.
(ii) / § £i n • • • n £},• n ■ • • n £5,, for each /'.
(4) Let {^31 ^3, } exhaust all minimal elements in {i/Q7 \/5«} f°i" rne
decomposition in (3). Then \fl = %\ D • • • n ^3,-.
(5) K, (I) = F, OP,) U • ■ • U \\(<p,.) is the irreducible decomposition of F+ (/).
3. Let X be a locally noetherian scheme, let f: X —> y be an affine morphism.
and let ^ be a quasicoherent (¾-Module. Follow the next steps to show that
H'iX.gr) = H'{Y.f,^) for each / > 0.
(1) Ri'f*&~ = (0) for each p > 0. (Let {K;};.eA be an affine open
covering of y such that f^](V/) is an affine open set of X for each /L Then
(Ri'f,^)^ = /?''(/,,j*.9v, for each/7 > 0 and X. By virtue of Theorem 3.23
(2). Ri'ify ),.^- = (0) for each p > 0 and /L Hence. /^/^ = (0) for each
p > 0.) ' '
(2) Applying the spectral sequence in (1.2.28)
£/>■</ = W\Y.Rl'f^) => H"{X.&-).
we have //''( y /^) ^ //''(A'. ^) for each p>0.
4. Let X and Y be separated algebraic schemes over a field k. and let /: A" —>
y be an affine morphism. By making use of Cech cohomologies. show that
H'(X.3~) = H''{Y. j\^) for each i > 0 for a quasicoherent ^-Module &.
5. Let R be a normal ring. Prove that R is a unique factorization domain if and
only if the divisor class group Cl(/?) vanishes.
6. Let X be a normal, projective algebraic variety, let f: X —> P^ be a morphism.
let 7i: PA —> P" be a projection, and let </? = n • y/: X —> P". Show that y/ is a
closed immersion if tp is a closed immersion.
7. Let /': X —> y be a surjective morphism between normal, projective algebraic
varieties, and let JS? be an invertible sheaf on Y. Verify the following assertions:
(1) f*5C is an invertible sheaf on X. Let il = {t/,},e/ be an open covering of
y which trivializes 2f. and let {0/,} be a family of transition functions of J?
with respect to il. Then /'"'(il) = {/_l(£//)}/e/ is an open covering of X which
trivializes f*5C. and {/*(0//)} is a family of transition functions of f*S? with
respect to /~'(il). where /*(()/:) is the image of 0/,- via the homomorphism of
the function fields ./'*: k(Y) ^ k(X).
(2) Obtain the injective homomorphism 2! <-> 2f ®fi, j\@x - /*(/*-S?) (cf.
Problem 1.2.9) by tensoring the injection of the structure sheaves ffy ^ f*@x
with 5C. Obtain from this injection an injective homomorphism /'*: H{)(Y.J^)
^-+ H°{X. f*2). Show that the last /* is an isomorphism provided (9Y = j\@x-
(3) Assume that all local rings of X and Y are unique factorization domains. Let

5. PROJECTIVE SCHEMES AND PROJECTIVE ALGEBRAIC VARIETIES 1?!
D be a Weil divisor on Y such that, with respect to an open coveringil = {t/,},e;
of Y. D is defined by an equation t/ = 0 on U,-. where t, ek(Y). Then show that
the local divisors (/*(?,)) on f~]{Uj) with /*(?,) G k(X) get patched together
to define a divisor ./'*£) on X. Furthermore, show that f*D ~ /*£>' if £> ~ £>'.
(We call /*£> the to to/ transform of £> by /.)
(4) With the assumptions of (3) above, show that f*D(n) = Z)(/'*(«■)) for each
o e#°(y.J5?). rr ^0.

CHAPTER 6
Nonsingular Algebraic Varieties
Let A be a ring. Consider a family of ideals {/,}„>(> of A satisfying the condition
that /„ C /,„ whenever n > m. For example, if we set /„ = /" for an ideal I.
the family {/"}„>o satisfies the condition. We can then define a topology on the
set A with an open basis {a + l„:a e A.n > 0}. Indeed, we have only to show
that if {a + /„) n {b + 1,,,) ~£ 0. for every c e (a + /„) n (b + 1,,,). there exists
r > 0 such that c + I, C (a + I,,) n {b + 1,,,). Write r = a + w = b + v with
m G /, and u e /,„. Take r so that r > max(H.m). Then c + I,- = (a + u + I,) C
« + /„ and r + /,. = b + v + I, C b + 1,,,. so c + /,. C (a + /„) n (b + 1,,,). With
respect to this topology on A. the addition and the multiplication of the ring A are
continuous mappings. In fact, for m < n. (a + I„) + (b + 1,,,) C (a + b) + 1,,, and
(a + /„) • (b + 1,,,) C ab + /„ + /„, + /„ • /,„ C ah + /„,. We call this topology the
linear topology defined by a family of ideals {/,},,>o-
An infinite sequence {x„}„>o of elements of A is called a Cauchy sequence if.
for each n > 0. there exists an integer jV = N(n) such that .v, — x, e /« for all
r.s > N. Define the sum and the product of two Cauchy sequences {x„}„>o and
{yn}n>o by {x„ +y„}n>o and {x„ •>'„}„>(). respectively. Then the sum and product are
also Cauchy sequences. We denote by C{A} the ring of all Cauchy sequences. Two.
Cauchy sequences {x„}„>o and {y„}„>o are cofinal with each other, by definition, if.
for each n > 0. there exists an integer N = N(n) such that xr —y,. e /„ for all r > N.
We denote it by {x„} ~ {y,,}- The set J consisting of all Cauchy sequences which
are cofinal with the zero sequence {0} is an ideal of C{A}. Let A = C{A}/J.
On the other hand, there exists the natural ring homomorphism /,„,,: A/l„ —»
/4//,,, whenever n > m. and /,,„• = /,„„ • /„,. if r > n > m. Namely, the set
{A/I„. f„„,;n > m > 0} is a projective system. So we may consider its limit lim /4//,,.
< n
which is identified with a subset {(x„)„>o;/m„Cx„) = ~x„, {n > in)} of the product
set n»>o A/I„. This subset is also a subring of the product of rings ]T„>o ^A'- We
define a ring homomorphism tp: A —»iim /4//,, as follows. Let {x„}„>o be a Cauchy
<—a
sequence. By definition, for each n > 0. there exists N(n) > 0 such that xr — x, G /,
for all r. s > N(n). We put y„ = xN^„y We may assume N(m) < N{n) if m < n. Then
{yn}n>o is a Cauchy sequence such that {>',,}„>o and {x„}„>Q are cofinal with each
other and that y„ —y,„ G /„, whenever m <n. So denoting by [{x,,}] the equivalence
class of {x„}„>o under the cofinality relation, we define y([{.v„}]) = (y„). where
y„ = >'„ (mod/,,). The cofinality class J is mapped by ip to the zero element of
lim /4//,,. ip is a ring homomorphism. If (x„) is an arbitrary element of lim /4//,,.
we have /„,„(-v„) = x,„ whenever w < «. If we choose x„ e /4 so that x„ = x„
(mod/,,), then x, — xA G /„ for all /■. .v > n. Hence, {x,,} is a Cauchy sequence. This
shows that <p is surjective. Similarly, we can show that if the cofinality class of a
133

134
11. SCHEMES AND ALGEBRAIC VARIETIES
Cauchy sequence {.v,,} is mapped to 0 by <p. then {.\-„} e J. So <p is injective as well.
This way we obtain the ring isomorphism <p: A ^> lim /4//,,.
< n
Let M be an /4-module. {/„M}„>o is a family of submodules of M such that
l„M C /„,A/ whenever m <n. As in the previous case for A. M is given a topology
with an open basis {£, + l„M\'i e M.n > 0}. and the addition and the
multiplication by elements of A in M are continuous mappings. We can also define Cauchy
sequences {£„}„>o and the abelian group C{M} of Cauchy sequences. Defining the
multiplication {.v,,} • {£„} by {.v„c„} for {.v,,} G C{A} and {£,„} e C{M}. we can
make C{M} a C{/4}-module. Furthermore, we can define the cofinality relation
on C{M} as in the previous case and set M = C{M}/JM. where Jm is the set
of Cauchy sequences in C{M} which are cofinal with the zero sequence {0}. Since
C{A}-Jm+J-C{M} C/m. m is regarded as an /4-module. As in the case for A. we
can also obtain an isomorphism of the abelian groups tpM '■ M ^> lim M/I„M. Let
(£,,) G lim M/I„M and (x„) G lim /4//,,. Noting that M/I„M is an /4//„-module.
< /; < n
we define the product (a„) • (£„) by (~x„ • £„). which is an element of lim M/I„M.
Thereby, lim M/I„M is a lim /4//„-module. The isomorphism ipM is compatible
< n < /;
with the ring isomorphism <pa{= ¥>)• Namely, we have <pm{x • £) = <p{*) • <Pm{£)
for a- G A and £ e M.
Define a ring homomorphism o: A —> A by u i—► [{«„}]. where u„ = a for each
n > 0. Similarly, we can define a homomorphism of modules oM: M —> A/ which is
compatible with /r. Namely. aM{a£,) = /t(u)<ta/(£) for « e ,4 and £ G A/. Clearly.
Kei"(T = n,i>o^« an(^ Ker(TM = f]n>0I„M. We often denote a (a) and om{£) by the
same notations « and £. respectively, where a £ A and £ G M.
Since {/„/4}„>o is a family of ideals of A. we can defined and the ring
homomorphism t: A —*A with respect to this family. An element of 1,,,A is represented
by a Cauchy sequence {«„}„>0 such that a„ G /„, for all n > 0. Let /> G p|,„>o^»^-
Then there exists a Cauchy sequence {«„" }„>o for each m > 0 such that {a„ }„>o
and {«„}„>() are cofinal Cauchy sequences for all r.s > 0. Consider then a Cauchy
sequence {«„' }„>o. which is cofinal with {«,," }„>o for each m > 0. On the other
hand. {«,,' }„>o- as a Cauchy sequence in C{A}. is cofinal with the zero sequence.
Hence. Kerr = f\iil>()l„,A = (0). Meanwhile, a Cauchy sequence {{«„ }„>o},„>o
of A is cofinal with a Cauchy sequence {{/>,, },,>0},„>0. where b,, — «„"' for all
n.m > 0. This implies that the ring homomorphism r: A —>/4 is an isomorphism.
Similarly, we obtain an isomorphism r« : M ^ M .
Let us consider the case where f)„>0I„ = (0). The linear topology on A then
satisfies the first axiom of separation "a.b e A. a ^ b implies that there exists some
n > 0. such that a £b +/„.b ¢. a + /„"'. Then we simply say that the linear topology
on A is separating or A is separated with respect to the linear topology. A Cauchy
sequence {«„}„>o in A is said to converge to an element a of A if {«„}„>o is cofinal
with the Cauchy sequence a (a) = {«}„>o. If the topology on A is separating, then
the element a is uniquely determined. Furthermore, since the ring homomorphism
a: A ^ A is injective. we can identify A with a subring of A by a. For a Cauchy
sequence {«„}„><) in A. {<t(«„)}„>o is a Cauchy sequence in A which converges to

6. NONS1NOULAR ALGF-BRAIC VARIFTU.S
1.15
the element {«„}„>o of A. In the case where p|H>()/„M = (0). we can show similar
results for M. We say that A and M are the completions of A and M. respectively.
Summarizing a part of the above observations, we obtain the following lemma.
Lkmma 6.1. Let A be a ring and let {/„}„>o be a family of ideals of A such that
hi Q hn whenever n > m. Let M be an A-module. Then we have:
(1) Let M he the completion of M with respect to {7„A/}„>(). Then M =
lim Mjl„M. There exists a natural homomorphism of A-modules gm : M —> M such
< n
that Ker<7A/ = f\n>QInM. M is an A-module. and <ta/ is compatible with the ring
homomorphism a : A —> A.
(2) If f\ll>0J„M = (0). M is separated with respect to the topology given by
{/„A/}„>(). Moreover. Im((TM) is a dense subset of M with respect to the topology
given by{InM}„>{). ^ ^
(3) M is separated with respect to the topology defined by {I„ M }«>o- andxM : M —>
A is tin isomorphism.
We shall prove the following result.
Lkmma 6.2. Let A be a noetherian ring. Let {/„}„>o be a family of ideals of A
such that, for every finitely generated A-module M and every A-submodule N of M.
there exists some p > 0 such that l„M flN = l„_p{lpM n N) for each n > p. Then
we have:
(1) The functor assigning to a finitely generated A-module M the A-module M is
an exact functor.
(2) For a finitely generated A-module M. there exists an isomorphism of A-modules
M ¢¢..1 A^M.
Proof. (1) For an exact sequence of ,4-modules
0 -► M, L M2 -^ M3 ->■ 0
we can define a sequence of ,4-modules in the natural fashion
M{ Lm2^ m?,.
Let {c„}«>o be a Cauchy sequence in M\. Suppose the Cauchy sequence {/ (c„)}„>o
in M2 is cofinal with the zero sequence. Namely, suppose that for each n > 0. there is
an integer N such that f {£,,-) e L,M2 for each r > N. By the hypothesis applied to M2
and a submodule /(Mt). there exists p > 0 such that I„M2 n f{M\) = l„-p{lrM2 n
/(M|)) for each « >p. Forn>p. we have / (£,.) € I„-p(IpM2n f(M\)) Cf{l„_nM\).
Hence. c,r e I„-,,M\ for each r. r 3> n > p. Namely. {£„}„>o is cofinal with
the zero sequence in M\. Next let {»„}„>(> be a Cauchy sequence in M2 such that
{g{n„)}„>o is cofinal with the zero sequence in M3. Namely, for each n > 0. there
is N = N(n) such that g(n,) e I„Mi = g{I„M2) each r > N. We may here assume
jV(1) < jV(2) < < N(n) < N(n + 1) < • • •. Then since {//„}„>o is cofinal with
a Cauchy sequence {nNl„<}„>o. we may replace {//„}„>o by {/7^(,0}«>(> and assume
from the beginning that for each n > 0. g(nr) G g(I„M2) whenever r > n. Then
for each n > 0. there is e„ e I„M2 and ^,, £ A/[ such that /(£„) = n„ - e„. Here
{n„ — £„}„>o is a Cauchy sequence in M2 which is cofinal with {n„}„>o- and {£„}„>o
is a Cauchy sequence in M\ by the hypothesis applied to M2 and f(M\). Hence.

1.%
11. SCHEMES AND ALGEBRAIC VARIETIES
Kerg = Im /'. We shall show that g is surjective. Let {C„}„>o be a Cauchy sequence
in M3. Replacing it by a Cauchy sequence in the same cofinality class, we may assume
that, for each n > 0. C,- - („ e /„ A/3 for each r > n. Choose >j0 e M2 so that #(/70) = Co
and nf] € A/2 for each r > 0 so that g(nlm) = £,• and /7,'°' - /70 G /()A/2. This choice
is possible because £,- — £0 £ /0A/3 = g(/o¥i). We obtain thus a sequence {/7, },>o
in A/t with /7(, = r\{). Next set /7I = /7J and choose /7,. ' for each r > 2 so that
#(?7,(l)) = C, and^1' -/7^ e I\M2 to obtain a sequence {/7,(1)},->i. Next, set /7^ = 1^
and choose /7, for each r > 3 so that #(/7,-) = C and /7, ' — /72 G /2A/2 to obtain
a sequence {/7, },>2- Repeat this choice inductively and obtain a sequence {/7, },>/
such that /7J0 = /7,-'_1). #(/7,(,)) = £,-. and /7,^ - /7)0 e /, A/2 for each r > i. Then if we
put /7,, = /7,, for each « > 0. the sequence {/7„}„>o is a Cauchy sequence in A/2.
In fact. J/,Ifi -»/„ =^+\ -^-^ =^, -^+^-^-0 G/„A/2. Furthermore.
g\ln)~ C«- Hence g is surjective.
(2) For an ^-module M. define a homomorphism of /i-modules hM : M®AA-*
M by (£. {a,,}) h-> {«„£}. where £ e M and {«„} is a Cauchy sequence in A. For
a homomorphism of ^-modules /: A/ —> N. we have /?/v • (f &> l.-j) = / • Am- I"
the case where F is a finitely generated free ,4-module, it is easy to show that hf is
an isomorphism. Since A is a noetherian ring, we have an exact sequence for every
finitely generated ^-module M:
Fx A F0 -^ M — 0.
where /¾ and i7; are finitely generated free ^-modules. One then obtains the following
commutative diagram:
Fx ®A A '/:x'j ; F0 ®A A Jl^lu M ®A A > 0
1"" J> iA"
F\ —-—> F2 —-—> A? > 0.
where the two rows are exact sequences and hft and h/.-, are isomorphisms. Then hM
is an isomorphism as well. □
Choose a proper ideal I of A and consider a family of ideals {/"}„>o. where
/" C /'" if n > m. The linear topology on A defined by this family of ideals is
called the I-adic topology. Similarly. {I"M}„>0 defines the Z-adic topology on an
^-module M, If A is a noetherian ring, the family of ideals {/"},,>(> satisfies the
hypothesis of Lemma 6.2 as shown in the following lemma.
Lemma 6.3. Let A be a noetherian ring, and let I be a proper ideal of A. Let M
be a finitely generated A-module. and let N be an A-submodule. Then there exists an
integer p > 0 such that I"M n N = l"~''(IrM n N) for each n > p. In particular.
IL = L. where L = On>0I"M.
Proof. Let R : = A © A/ be a direct sum of /4-modules. Define the multiplication
on R by (a.x) • (h.y) — (ab.bx + ay). Then (1.0) is the identity element and M.
identified with (0) © M. satisfies M2 = (0). R is thus a noetherian ring (compare
Problem II.3.1). and A/ and N. identified respectively with (0) © A/ and (0) CD N.
are ideals of R. Furthermore, if we set a = I © A/, a is an ideal of R. By virtue of
Lemma of Artin-Rees (1.1.31). there is p > 0 such that for each n > p. a"M C\N =

6. NONS1NGULAR ALGEBRAIC VARIETIES
1.17
a" p(apM n N). By a straightforward computation, a"M n N = I"M n N and
a"-i'(a>'M n /V) = I"-i'(I''M n AT). By (1.1.31) again, we have IL = L. □
The following are the results we often refer to in the present book.
Theorem 6.4. Let (/1. m) At' a noetherhin local ring, and let A be the completion
of A with respect to the m-odie topology (the m-adic-completion of A. for short). Then
we hare:
(1) A is separated with respect to the m-adic topology.
(2) (A.mA) is a local ring, and A/mA = A/m.
(3) A is a flat A-module.
(4) M = M <S>a A for a finitely generated A-module.
Proof. (1) By Lemma 6.3. rru/ = J. where J = n„>om"- Then J = (0) by
Nakayama's lemma (1.1.28). Hence. Lemma 6.1 shows that A is separated with
respect to the m-adic topology.
(2) (A.mA) is a local ring if and only if every element of A — mA is invertible.
(Prove this as an exercise.) Take an element .v of A — mA. which is represented by a
Cauchy sequence {«„}„>o in A. where we may assume a„^\ — u„ e m". Hence. a0 ¢. m.
Then a„ ¢. m for each n. and {ufl}„>o is easily seen to be a Cauchy sequence in
A. This sequence converges to x~{. It is apparent that the ring homomorphism
a: A —> A induces a homomorphism of fields a: A/mA —> A/mA. On the other
hand, if {«„}„>o is a Cauchy sequence in A. a„ (modm) e A/mA (n ^> 0) is
determined independently of n. So. we obtain a homomorphism A/mA —> A/mA
which establishes, together with a. an isomorphism A/m = A/mA.
(3) A is a flat ,4-module if and only if j ®A A: J ®A A —* A <$>A A = A is an
injection for an ideal J of A and the natural inclusion j : J —> A. (Try to prove this
as an exercise. Compare [6. Theorem 7.7].) If we note that A is a noetherian ring.
then this assertion together with the next (4) follow from Lemma 6.2. □
Example 6.5. Let A = k[.\\ v„] be a polynomial ring in n variables over k
and let I = (x\ .v„). The Z-adic topology is separating. (In fact. IL = L by
Lemma 6.3. where L — f]„>o^"- L 's a nn'tely generated ideal of A. In view of
the proof of Nakayama's lemma (1.1.28). dL = 0 for d e A. d ^ 0. Since A is an
integral domain, we conclude L = (0).) The completion A of A with respect to the
Z-adic topology is a formal power series ring in n variables over k. which is the set of
(formal) infinite series J2('h---i„x'\' •••*,'," (<V-/„ e k'- 'i '"» > 0) with coefficients
in k subject to the addition and the multiplication as defined below
(X>,-,-vi'
(X/'''-'»-Yi
Replacing the field k by a ring R. we can also define a formal power series
ring R[[x\ a,,]] with coefficient ring R. Let (5". m) be the localization of A with
respect to I. Then the m-adic completion of 5" coincides with A. We have also
•■•.vi').(^rf(V.,„.vJ'...^)
= E E c)r-iAr-k„ X'l ■■••<".

138
II. SCHEMES AND ALGEBRAIC VARIETIES
k[[x\ v,,]] = k[[.\t a-„_|]][[.y„]]. The ring k[[x\ x„]] is a noetherian ring
by the following lemma.
Lemma 6.6. Let R be a noethcriun ring. Then /?[[.v]] is a noetherian ring.
Proof. An element g of R[[x]] is expressed as a formal power series
g = al/x'1 +a<lnx'l + i+--- . a,, ^0.
where we call udxi} the term of the lowest degree of g and denote this term by LF(g):
we also write d = deg#. Clearly. LF{g) is an element of the polynomial ring R[x].
Now let I be an ideal of R[[x]]. and let /* = {LF(g):g el}. Let a be the ideal
of R[x] generated by /*. Since R[x] is a noetherian ring, there are finitely many
elements £1 £,„ of /* such that a = J2" i ^[A'K;- F°r eacn '■ £/ = LF{g;) with
g,,- e /. We shall show that I = Yl"-\ ^ [[-*]]#;• Choose any element g of I. Since
LF(g) e /* C a. we have LF(g) = ^" , £,/2,0 with homogeneous elements h,o of
R[x] (\ < i <n). Then deg{g - £"_ , £,/i,0) > degg. Since g - £"M £,½) € 7- we
find homogeneous elements /2,1 (1 < i < n) by the above argument such that
deg \g - 2^ gi (/i,-() + A,,) > deg h? - ^ g, /7,0 . deg A,-1 > deg /½ (for each /).
(-1
Repeating this argument, we find homogeneous elements {h/,; 1 < i < n. j > 0} such
that
deg hi j > deg/!,,_! (for each /)
and
deg (g - j^£;(A/0 + /2,1+--- + A,-,-) J
> deg Ig-f^giihio+hn +---+^7-1)] (for all 7 > 1).
Then h, := 5Z,>o^'/ 's an element of /?[[.v]] and g = ^" , gjh:. D
Corollary 6.7. Ltj/ (/?.m) he a noetherian local ring, and let R he the m-adic
completion of R. Then the following assertions hold:
(1) Let ¢1 ¢,, be elements of m such that the residue classes £, ¢,, o/
¢1 ¢,, modulo m2 constitute a basis of the finite-dimensional vector space m/m2
over the field R/m. Then m = (£| £„).
(2) Suppose R is an algebra over a field k and the natural homomorphism k —>
/? -^- R/m induces an isomorphism between k and R/m: we then say that R contains the
residue field. Then there exists a surjective homomorphism of rings p: k[[xi v„]] —»
R from the formal power series ring to R. In particular. R is a noetherian ring.
Prooe. (1) Let n = ^" , /?£,■. The m = n + m2. Let M = m/n which is a
finitely generated /^-module and satisfies the condition mA/ = M. So M = (0) by
Nakayama's lemma. Hence, m = n.
(2) Note that R is a subring of R. Define p: k[[x{ v,,]] -+ R by f =
H/>()//'(-vi -v«)^ S/x)//'(^1 £«)• I* 's readily shown that/; is well defined
and that p is a ring homomorphism. We shall show that p is surjective. For each /- > 0
and every element />, of m'. there exists a homogeneous polynomial /,-(xi x„)

6. NONSINGULAR ALOt-BRAIC" VARIUTIHS
139
of degree r and with coefficients in k such that br — f ,{c,\ C,) 6 m'+l. Every
element of R is represented by a Cauchy sequence {«,■};■ >o with «, + i — ar e m' + 1 for
each r > 0. Then choose /,-(a'i v,) e k[x\ a,,],, so that
a,- - ^//(^i C"») e m'"1 (for each r > 0).
/-()
Set / = ^2j>0fj. Then the Cauchy sequence {«,},>o converges to /;(/) in R. So
p is surjective. □
The next result is called the altitude theorem of Krull. We refer to [7. p. 26] for
a proof, though the proof is not hard.
Lemma 6.8. Let R be a noethcrian ring. Let I be an ideal of R which is generated
by n elements. Then every minimal prime divisor p of I has ht(p) < n. In particular.
if(R.m) is a noetherian local ring, we have dim/? < dimR/m(m/m2). (Henceforth, we
denote K-dimR also by dim/?.)
The next result is a significant one in the theory of commutative algebras.
Lemma 6.9. Let (R. m) be a noethcrian local ring with d — dim R > 0. Then
there exist d elements u\ a(/ of m such that the ideal [a\. ■ ■ ■ . a,/) is a primary
ideal belonging to m.
Proof. Let p0i po,(1 exhaust minimal prime ideals of R. Then m ^ |J'."_, p(),.
(In general, for prime ideals p[ p, and p. p, ¢. p; (/ f j) and p C |j' , p, imply
p Q Pi (for some i). In fact, if p ¢, p, for each i. pnp] • • -np, n- • -Dp, ¢, p, for each /'.
So there is some .v,■ G p n pi n • • • n p,- n • • • fl p, and x ¢. p;. Set v = x\ H V x,. Then
y ¢- U/_i Pi andy e P- ^n's 's absurd.) Let a be an element of m. not in (J'.'^_, p(),. and
let pn pi,, exhaust minimal prime divisors of a\R. Then ht(pw) = 1 for each
/". In fact, ht(pi,) < 1 by the altitude theorem. If ht(pi,) = 0. then pi, is a minimal
prime ideal. This contradicts the hypothesis a ¢. \J'"_l po/- For s < d. suppose we
have chosen elements u\ a, in such a way that ht(pw) = s for any one of minimal
prime divisors p,i p,(> of the ideal (a \ a,). We choose an element «A + i ofm.
not in \J'j'_i pM. (If m = (J^, pv,. then m = pv (for some j) by the above reasoning.
and this contradicts the hypothesis that d = ht(m).) Let ps + u p.<, + \.,\ 11 exhaust
minimal prime divisors of the ideal («[ «,n). Then ht(piS + i.,-) = .v + 1 for each
/'. In fact. ht(p.s-11.,-) < .v + 1 by the altitude theorem. Since (a\ a,) C p., + ),. we
have p,4+ 1.,- 2 Pv./ for some j. Hence. p, + i., > .v. If ht(pv+i.,) = x. then p, + i., = p,.;.
which contradicts the choice of ai + i. Thus, if we choose a\ a(j inductively as
above, then m is the unique minimal prime divisor of the ideal («i «,/). Hence.
if we consider a primary ideal decomposition of ( a i a(j). we see that (a \ a(\)
is a primary ideal belonging to m. □
Definition 6.10. Let (/?.m) be a noetherian local ring. A collection {a\
a(i) of elements of m is called a system of parameters of R if d = K-d\mR and
m = \/(«i a,/). By Lemma 6.9. there exists a system of parameters. If we can
choose a system of parameters {ti\ alt} so that m = (ci\ atl). we call R a
regular local ring, and we call {u\ a,/} a regular system of parameters of R.
By a result of Auslander (Lemma 5.16). a regular local ring is a unique
factorization domain, and hence, a normal ring.

140
II. SCHhMFS AND ALGEBRAIC VARIETIES
Lemma 6.11. Let (R.m) be a noetherian heal domain, and let R be the m-adie
completion of R. If R is noetherian (this is. in fact, always the case [7. (17.6)]). we
have K-dimfl = K-dimfl.
Proof. Our proof proceeds by induction on d : = K-dim/?. If d = 0. the zero
ideal (0) is a primary ideal belonging to m. Hence, m" = (0) for some N > 0. So the
m-adic topology on R is nothing but the discrete topology. So R = R. Suppose d > 0.
and let {a\ a,/} be a system of parameters of R. Then m" C I : = (a\ att)
for some N > 0. Since m = mR (Theorem 6.4). we have m" C IR. So by the
altitude theorem of Krull. we have K-dim/? < d. Next note that a\ is not a zero
divisor of R. In fact. a\\ R-+ R (x ^> a\x) is an injection because R is an integral
domain. Since R is a flat /^-module (Theorem 6.4). we infer that a\: R —> R is
an injection as well. Hence. a\ is a nonzero divisor of R. This implies that every
minimal prime divisor of a\R has height 1. (In fact, if ^3 is a minimal prime divisor
of a\R with ht(«p) = 0. then «p = ((0): x) (1.1.23). Hence. ax becomes a zero
divisor which is a contradiction.) It follows that K-d\mR/a\R < K-dim^ - 1.
Suppose Y^-d\mR/a\R < K-dim/? — 2. Then we can choose elements «2 as
(x = ¥^-d\mR/ a\R + 1) of m so that their residue classes modulo a\R constitute
a system of parameters of R/a\R. The ideal («1. iii a,) is then a primary ideal
belonging to rh. By virtue of the altitude theorem, we have K- dim R < s while
v = K- dim R/a\R + 1 < K-dim/? - 1 by the hypothesis. This is a contradiction. So
we know that K-d\mR/a\R — K-dimR - 1. By means of Theorem 6.4 (4). R/a\R
is the (m/a\R)-'ddic completion of R/a\R. We can show by the argument above
that K-dimR/a\R = d — 1. The induction hypothesis implies that K-dim R/a\R =
K-dim R/a\R = d - 1. Hence. K-dim^ = K-dim ft. □
As a conclusion of the above arguments, we obtain the following result.
Thkorem 6.12. Let (R.m) be a noetherian local domain such that R contains the
residue field k = R/m and d = AT-dim R. Then the following eonditions are equivalent
to each other:
(1) R is a regular local ring.
(2) R=k[[x\ v,,]].
Proof. (1) implies (2). Let {«1 a,/} be a regular system of parameters of R.
By Corollary 6.7 there exists a surjective ring homomorphism p: k[[x\ x,/]] -+ R
such that p(xi) = a,-(1 < i < d). In particular. R is a noetherian ring. By
Lemma 6.11. R and A:[[.V| \,/]] have the same Krull dimension d. So /; must
be an injection.
(2) implies (1). R/mR = k. and m/m2 = mR/(mR)2 as ^-vector spaces. Hence,
dim/, m/m2 = d. By virtue of Corollary 6.7 (1). we know that R has a regular system
of parameters. □
We now consider the case where (R.m) is a local ring of an algebraic variety.
Let X be an irreducible reduced algebraic scheme denned over a field k. let P be a
/^-rational point of X. let R = @x.? and let m = m^.p. R is then a noetherian local
domain which contains the residue field k. We say that P is a nonsingular (or smooth)
point of X if R is a regular local ring, and a singular point of X otherwise. If k is
an algebraically closed field, then X is an algebraic variety defined over k and every
closed point of X is a /c-rational point. If every closed point is a nonsingular point.

6. NONS1NGULAR ALGEBRAIC VARIETIES
141
then we say that X is a nonsingular algebraic variety. Otherwise, we set SingX =
{P 6 X(k); P is a singular point of X} and call it the singular locus of X. We quote
the next result without giving a proof (cf. [7. (28.3)]).
Lemma 6.13. Let (R.m) be a regular local ring, and let p be a prime ideal. Then
Rv is a regular local ring.
Let us go back again to the set-up before Lemma 6.13. Replacing X by an affine
open neighborhood of P. we assume that X is a closed subscheme of the affine space
A"k. Choose the coordinate functions x\,.... x„ of A", so that the point P is given
by (0 0). Set & = @k».p- We have the natural surjective ring homomorphism
p: 3 = k[[xi x„]] -► R -► 0
which is induced by the natural ring homomorphism
a:k[Xi xn\^Y{X.@x).
So if we set I = Kerrr. then we have Ker/? = 1&. Let {/,■; 1 < i < N} be a system
of generators of/. Since /,-(0 0) = 0 for each i, /,-. viewed as an element of &,
is expressed as
n
fi = 2_,anxj + (terms of degree > 2).
/=i
We let M the (N x «)-matrix (a,-/).
Lemma 6.14. If r = rankM, there exists a surjective ring homomorphism from a
formal power series ring to R
n:k[\y, y„_r\\ -+ R -+ 0.
Let d = dim/?. If r = n — d. then R is a regular local ring. Conversely, if R is a
regular local ring, we have r = n — d.
Proof. After a suitable change of indices, we may assume that S := (a,-7)i<,^-<,
is an invertible matrix. Then replacing '(/i /,) by (S~])'(f\ /,.), we may
assume that S is the identity matrix. Let y, = x,+, (1 < i < n - r) and >>„_,•+,• = /,-
(1 < i < r). Then we have y„-,■+, = x,+ (terms of degree > 2) for 1 < i < r. These
relations entail
x, - yn-r+i - gi e ((x, xn)@)c+l
for an arbitrary integer c ^> 0. where gt is a polynomial in y\ y„ and g, = gn +
• • • +gn- with gij the homogeneous part of gj of degree j. Hence. & = k[\y\ y„]].
In particular. W — {y„-r+i y„. fr+\ fn)@■ So we obtain the natural
surjective ring homomorphism n: k[[y] yn-r]\ —> R- Let n = {x\ x„)&. The
existence of n implies n — r > d. If r < n — d. then the above argument implies
dim^ njW + n2 = n-r>d. Meanwhile, m/m2 is isomorphic to njW + n2. Hence,
dim/t m/m2 > d. So R does not have a regular system of parameters. So r > n - d
provided R is a regular local ring. Therefore, r = n - d. O

142
II. SCHEMES AND ALGEBRAIC VARIETIES
If we regard /,• (1 < i < N) as the elements of k[x\ x„], then the matrix
M = (a,;) is equal to the following matrix:
/<£(„ ... ^1,,,\
ox i ax„
M
^(/>) ... dfN (P)
\ dx\ dxn '
where j£-(P) is the partial derivative of /,■ with respect to x,- evaluated at the point
P. Let Q~{c\ c„) be another /c-rational point of X. Then {x\—c\ x„ — c„}
is a regular system of parameters of $\«.q. and the following is an exact sequence
0 ^ Wk:q -> it,:Q ^ 0x.Q -+ o.
where &a".q — k[[z\ ■ ■ • •, "«]] (z/ = xi — c,-). The system of generators {/,•; 1 < i < N}
of I is expressed as {g,-; 1 < i < N} after the change of variables gi{:\, zn) =
f,-(z\ + c\ z„ + c„). Consider then the (N x n) matrix (-^-(Q)). By a simple
computation, we have
dgj dfi{:x+cx....,:„+c„) df,
- - -{z] +cll....zn +c„).
where
dzj dzj dxj
^-(0 0) = -z—\c\ (-■„) = -^-(2).
ozj axj oxj
Hence, <$x.q is a regular local ring if and only if
Let J be the ideal of k[x\ , x„] generated by all (« — d) x (n — d)-minors of the
(N x «)-matrix (g~). If k is an algebraically closed field, then the set of singular
points (called the singular locus) of X is a closed subscheme of X defined by the
ideal I + J. Summarizing the above observations, we obtain the following result.
Theorem 6.15. Let X be an algebraic variety of dimension d defined over an
algebraically closed field k. Then the following assertions hold:
(1) Suppose X is defined by an ideal (f\ f m) as a closed subscheme of the
ajfine space A", = Spec k[x\ x„]. Then a closed point P of X is a nonsingular point
if and only if rank((-g^-(P)) = n — d.
(2) The singular locus SingX of X is a closed subset. Furthermore, Sing(X)rect =
{P G X; ®x.p is not a regular local ring} as a closed subscheme. Hence codim^ Sing(X)
> 2 provided X is a normal algebraic variety.
We omit the proof of the latter half of the assertion (2). If X is normal the local
ring @x.e, °f a point £ of codimension 1 is a discrete valuation ring, so £ ¢. Sing(X)red.
Hence, codim^- Sing(A') > 2. The result given in the assertion (1) is called the
Jacobian criterion of nonsingularity.
Hereafter in the present chapter, we shall define the sheaf of differential forms
Qx/k attached to an algebraic variety X and discuss the nonsingularity of X in terms
of Qxjk- Let B be a ring, and let A be a 5-algebra. Let M be an ,4-module. A

6. NONS1NGULAR ALGEBRAIC VARIETIES
143
homomorphism 3: A —> M of B-modules is called a B-derivation of A (with values
in M) if it satisfies the condition that
8(xy) = xS{y) + yd{x), for all x.y G A.
This condition implies 8(b) = 0 for each b G 5. We denote by Der#(<4.M) the
set of 5-derivations of /1 with values in M. If we define a8 + a'8': A —> M by
(«<5 + «'<5')(x) = «(5(x) + «'<5'(x) for 8.8' e DerB(A.M) and «V G /4, then we
have «<5 + «'£' 6 DerB(/l. M). With this operation, DerB(^, M) is an ,4-module.
For an /4-module homomorphism /: M —> A7", define an /i-module homomorphism
/,: Der^.M) -»DerB(^.^) by/,(<J)U) = f(5(x)) (8 G DerB(A. M), x e ^).
Clearly, (g • /)* = g* • /* for /: M —> A7 and g: Af —> P. and (1m)* is the
identity morphism. In other words, the correspondence M >-+ Der b(A. M) is a
covariant functor from the category of ^4-modules into itself. We shall show that
there exist an <4-module (denoted by £Ia/b) and a B-derivation d: A —> Q.a/b sucn
that HomA(Q.A/B- M) —f DerB(A. M)(f \-+ f • d) for every ^4-module M. The
.4-module £1,4/b is constructed as follows:
The 5-algebra A ®>b A is regarded as an ^4-module via a(x ® y) = «x <g> >'
((V.x.j G /4). The multiplication //: /1 ®# A -+ A. /i(x ® y) = xy is then an A-
module homomorphism. Set I = Ker/z. We have an exact sequence
0^1 ^A®bA-^A^0.
Since I is an ideal of A ®B A. I /I2 is defined as an /i-module.
Lemma 6.16. (1) The ideal I is generated as an A-module by {a®\ — \®a;a &A}.
(2) Set da = a <g> 1 — 1 ® a (mod/2). 77;e« ?/?e A-module I /I2 is generated by
{da; a G ^4}. 7%e mapping d: A -+ I /I2, a >-+ da, is a B-derivation of A.
(3) For an A-module M, define <p: Hom.4 (///2. M) —> Der#(<4. M) by f >-+ f • d.
Then <p is an A-isomorphism.
Proof. (1) We have JZ, «<" ® «,' ^ / if and only if £\ a,a' = 0. Hence, £\ «, ®
«/ = -£,«/«®l " 1®«,')-
(2) By (1). we know that the ^4-module ///2 is generated by {da;a G A}. We
have
d(aa') = a a' <8> 1 — 1 ® <y«'(mod/~)
= «(«' ® 1 - 1 (gi a')
+ a\a ®\ -\ ® a) - (a ®\ ~\® a)(a' ® 1 - 1 ® «') (mod/2)
= ada' + a'da.
d{ba) =ba ®\ -\®ba (mod/2)
= i(« <g> 1 - 1 <g> a) (mod/2)
= W« (Ae5).
Hence, d is a 5-derivation.
(3) Suppose /-<i=0 for / G Hom/1(///2. M). Since {da\a G ^} generates
the /i-module ///2. we have / = 0. Hence, ip is injective. Given 8 G Der#(/1.M).
define /: /1 ®B /1 -> M by /(« ® a') = -a8(a'). Since /(a ® 1 - 1 ® a) = 8(a)
and f((a® 1 - 1 ® «)(«'«> 1 - 1 ®a')) = a<5(«') + «'(5(«) -5(««') = 0. /|/ induces

144
11. SCHEMES AND ALGEBRAIC VARIETIES
an A-mod\i\e homomorphism f: I j' I2 —> M such that 8 = f • d. Therefore, <p is
surjective. □
Denote the ^4-module I /I2 by £Ia/b and call it the module of differential forms
(or simply the differential module) of the 5-algebra A. Qa/b possesses the properties
as listed in the following lemma.
Lemma 6.17. (1) Let B' be a B-algebra, and let A' = A ®B B'. If we regard
A' as a B'-algebra in the natural fashion, then we have an A'-module isomorphism
Qa'/b' — &a/b ®b B', where £Ia/b ®b B' is viewed as an A'-module via (a <g> b'){8 ®
b") = ad®b'b" {aeA.de QA/B-b'.b" e B').
(2) Let S be a multiplicatively elosed set of A. Then there is an S~] A-module
isomorphism Q^-'a/b — S~]Qa/b-
(3) Let C be a ring. Suppose B is a C-algebra. Then there exists the following
natural, exact sequence of A-modules:
(*) £lB/c ®b A^ QA/C -► ClA/B ->• 0.
(4) Let J be an ideal of A. and let R = A/J. Then there exists the following
natural, exact sequence of R-modules:
(**) JI J1 ^ QA/B ®A R -> &R/B -> o.
(5) Q.a/b is a finitely generated A-module provided A is a finitely generated B-
algebra.
Proof. (1) Note that we have an isomorphism Hom^^Q^/^ ®B B'.M') a
Hom/!(Q/J/B. M') for an ^'-module M'. Then we obtain an ^'-module isomorphism
Qa'/b' — Qa/b ®b B' as required if we show that the natural ^'-module
homomorphism a: DerB(A. M') —> DevBi{A'. M') is an isomorphism, where a (8) is defined
by a(d)(a ® b') = (1 <8> b')S(a) for 8 e DerB(A.M'). It is readily shown that a
is injective. Given 8' e Ders'(A'. M'), define a 5-homomorphism 8: A —> M' by
8(a) = 5'{a cg> 1) {a e A). Then 8 6 DerB(A.M') and 8' = a(8). Hence, a is
surjective. So a is an isomorphism.
(2) Let M' be an 5~'^-module. Define an S-1,4-homomorphism /?:
Der B(A.M') -> DerB(S-lA. M') by B{S)(a/s) = {sS{a) - aS{s))/s2 for 8 e
DerB(A. M'). We have only to prove that /? is an isomorphism. The proof is easy,
so we omit it.
(3) For an A-modu\e M. we can define a sequence of <4-modules in the natural
fashion.
(*)' O^DerB(A.M)^Derc(A.M)^Derc(B.M).
Namely, p maps a 5-derivation 8: A —> M to the same derivation viewed naturally as
a C-derivation. and a maps a C-derivation 8: A —> M to the restriction 8\B. Then we
can readily show that (*)' is an exact sequence. The sequence (*)' gives rise to an exact
sequence (*). For example, if we identify DerB(A.QA/B) with HomA(QA/B.QA/B)
and Derc(AMA/B) HomA{QA/c.QA/B). then p{lnA/B): QA/c -> &a/b is the A-
homomorphism considered in the sequence (*). Similarly, the ^4-homomorphism
(t(1q4/( ): QB/C®BA^QA/c is the one considered in the sequence (*). The exactness
of the sequence (*) follows from the fact that the sequence (*)' is exact for an arbitrary
A -module M. The proof is left to the reader as an exercise.

6. N0NS1NGULAR ALGEBRAIC VARIETIES
145
(4) Taking the composite of the ^-homomorphism d: A -+ Q.A/B with the natural
homomorphisms, we obtain
df:J^A^ Q.A/B -► QA/B ®A R
such that djR(J2) = (0). So we get an .K-homomorphism d: J/J2 —> £lA/B ®a R- For
an /^-module M. d : HomA(Q.A/B.M) —> Hoiiir(///2.M) is given as d {S)(a) =
S(a) (<5 6 DerB{A. M), a = a (mod/2), a e J), where HomA(QA/B. M) is identified
with DerB(A. M). In order to prove that the sequence (**) is exact, we have only
to show that the following sequence of ^-modules
(**)' 0->Der*(/?.M) -^ DtrB(A. M) L HomR{J/J2. M)
is exact for every ^-module M. where r is given as 5 ^> 8 ■ n with the natural residue
homomorphism n: A —> R. It is easy to show that the sequence (**)' is exact.
(5) By hypothesis, we can express A as the residue ring of a polynomial ring
over B.A ~ B[x\ x„]/J. Note that QB[Xl x„]/b is generated by dx\ dx„
as a B[x\ x„]-module. The exact sequence in the assertion (4) then asserts that
Qa/b is a finitely generated /i-module. □
Let f: X —> Y be a morphism of schemes. Let j be a point of Y. let V = Spec(5)
be an affine open neighborhood of y. and let it = {Ux}xeA {Ux = SpecO^)) be an
affine open covering of /_l (V). Since Ax is a 5-algebra. we can define the differential
module £Ia,/b- We denote by £lu;/v the quasicoherent Module {QA//B)~ on Ux- For
A.// 6 A, let U = Spec(A) c Ux fl UM be an affine open set. Consider quasi-coherent
Modules Q(7/r = {QA/B)~ and £lu,/v\u on U. There is a 5-algebra homomorphism
Ax —► A. which induces an ^4-module homomorphism QA)/B ®A} A —> QA/B, and
hence, a homomorphism of @u-Modules Ox'. Qu,/v\u —> Qu/v- If everything is
restricted on an affine open set D(h) of C/(c Ux). then Q^/rl/H/,) = ^ u,-']/b an^
^/i-lfl(A) = (^[A-']//?)~. where Ax[h~l] = A[h~1]. Hence. Ox\u (and 0;.) is an
isomorphism. Thus, there is a natural isomorphism Qujvlu^nUf, — ^c/„/r\u,r\L't,
(for each A.// 6 A). This way. the {QL,jV}xeA get patched together to form a
quasicoherent (¾-Module 0.x/y- We call this (f>-Module the sheaf of differential
forms attached to the morphism of schemes f. We obtain the following lemma as
a corollary of Lemma 6.17.
Lemma 6.18. (1) Let q : Y' —> Y be a Y-scheme, let X' = X x Y Y'. and let
q' = X xY q. Then Qx'/Y' = <i'*Qx/Y-
(2) For an open set U of X. we have Qu/y — Qx/y\u-
(3) Let f : X —> Y and g: Y —> Z be morphisms of schemes. Then we have the
following natural exact sequence of quasi-coherent ®x-Modules:
f*Ox/Y —> Q-x/z —>■ Ox/y —> 0.
(4) Let F be a closed subscheme of X, and let J" be the defining Ideal of F. Then
we have the following exact sequence of quasicoherent &x-Modules:
SjS —> QX/y\f —>■ Qf/y -^ 0.
(5) Let f: X —> Y be a finitely generated morphism. Then Qx/y "' a finitely
generated ®x-Module. In particular, if X is a noetherian scheme. Qx/y w a coherent
Module.

46
II. SCHEMES AND ALGEBRAIC VARIETIES
Proof. (1) Let U = Spec(A). V = Spec(5). and V = Spec(5') be affine open
sets of X. Y. and Y'. respectively, such that U C f~\V) and V C q~x{V). and
let U' = U xy V. ThenQY7K,|[;, = {Q,A./B^~ and q'*nx/Y\u' = {^a/b ®b #T-
where /4' = A ®B 5'. By Lemma 6.17, Q.A'/B> — ^/« ®# 5'. Since X' is covered
by affine open sets U' of this kind, we have an isomorphism Qx'/Y' — o'*Qx/y-
(2) Let W = Spec(/i) be an affine open set of U such that f(W) is contained
in an affine open set Spec(5) of Y. Then the restrictions of Ql/k ar>d Qj/k|c onto
W7 coincide with (Qa/b)~.
(3) Let C/ = Spec(^). V = Spec(5). and W = Spec(C) be affine open sets of
X.Y. and Z. respectively, such that U C f~x{V) and F C g~'( W). By virtue of
Lemma 6.17. we have an exact sequence of /Fmodules
&b/c ®s A —> ^/c —► tl^j/B ->• 0.
This yields an exact sequence of ^'-Modules
(/ \u)*Qv/w -^ Qu/w —> Qi'/v ~* 0.
Vary U on affine open sets of f~[(V) with V and W7 fixed, and then vary V on
affine open sets of q~l(l¥) with W fixed, in order to patch together QLyr.Q6yn/.
and (/|j./)*Qi'/^. The homomorphisms y> and (// are also patched up under these
patching processes. Thus, we obtain the exact sequence in the assertion (3).
(4) By means of Lemma 6.17 (4). we patch up the exact sequences obtained
locally, and thus obtain the exact sequence as required.
(5) £Ix/y 's a quasicoherent ^y-Module by the construction. If f: X —> Y is a
finitely generated morphism. then Qx/y is a finitely generated ^--Module by Lemma
6.17 (5). □
Lemma 6.19. Let X be an irreducible, reduced algebraic scheme of dimension n
defined over k. Then the following assertions hold:
(1) For a closed point x of X. put & = @x.x* m = mx.x- und k(x) = &/m. If
k(x) is separable over k, we have
dimA.(v) Q^/t.v ®ff k{x) < dimA.(v) m/m2.
where 0.x/k.x « the stalk ofQ.X/k at x.
(2) Let K = k(X) be the function field of X over k. IfK/k is a separable extension,
then we have
n < dimA.(v) £lX/k.s ®* k{x).
Proof. (1) Let L/k be an algebraic extension of fields. We shall show that L/k
is separable if and only if Cl/./k = (0).
"Only if" part. It suffices to show that Derk(L.L) = (0). Let <5 e Derk(L. L).
Let I be an element of L. and let /(-) be the minimal equation of £ over k. Since
f is separable over k. /'(£) f 0. Meanwhile. f'{£)5{£) = 3{f{£)) - 0. Hence.
<J(£) = 0. So 5 = 0.
"If part. Suppose L/k is not separable. Then there are an intermediate field
M/k and an a e M such that L^ M and L = M(axlp). where p is the characteristic
of k. Let £ = tv1/''. Then it is easy to show that DsrM(L.L) ^ (0). For example.

6. NONS1NGULAR ALGEBRAIC VARIETIES
147
write L = M[t]/(tp - a). Then the M-derivation d/dt on M[t] induces a nontrivial
M-derivation 5 on L. (In fact. S(£) = 1.) The exact sequence
0 -» DerM (L. L) -» DerA (L. L) -> DerA (M. L)
then implies Derk(L. L) ^= (0). Here we apply Lemma 6.17 (2) to k(x) = <$/m to
obtain an exact sequence of £(x)-modules
m/m2 -► Qff/k ®g k{x) -► Qa.(v)/a- ->• 0.
where 0/((v)/a. = (0) by virtue of the hypothesis that k(x)/k is separable and the
above remark. Since Q-x/k.x ®ff k(x) = Q.&/k ®ff k(x) by Lemma 6.17 (2), the exact
sequence yields the inequality as required.
(2) Before starting our proof of the assertion, we shall make the following remark.
Let A be a 5-algebra and let M be an /1-module. Define the multiplication law in
a direct sum A © M of <4-modules by {a. u)(a'. u') — (aa'.au' + a'u), which makes
A ® M an ^4-algebra with the identity element (1.0). A is regarded as a subalgebra
of A © M via the embedding A —> A © M.a >-> (a, 0). Hence, A © M is identified
with A + Me by the mapping (a.m) *-^> a + me. where we formally introduce an
element e with e2 = 0. Then A = A + Mej{e). where (e) = Me. Consider the
natural residue homomorphism n: A + Me —> A. a + me >-> a. We call a 5-algebra
homomorphism A: A —> A + Me a B-lifting of the identity mapping \A q{ A if A
satisfies the condition n • A = 1^. Then we can easily verify that there is a one-to-
one correspondence between DerB(A. M) and the set {A; 5-lifting of 1^} given by
8 h-> AA, where A#(a) = a +S(a)e.
Now let AT = k(X). Since AT/£ is a separable extension by the hypothesis, there
exists a transcendence basis {x\ x„} of K jk such that AT is a separable algebraic
extension of Kq = k(x\ x„). By the remark given in (1). the exact sequence in
Lemma 6.17 (3) gives rise to the following surjective homomorphism
&Ko/k ®Ka K -► QK/k -► 0.
We shall show that this is indeed an isomorphism. For this, it suffices to show that
the following injective homomorphism
0 -> DerA.(A\ K) -> DerA.(A"0. K)
is surjective. Let Sq € Der^ATo- AT). By the above remark, <?o corresponds to a A>
lifting Ao: K0 —> AT0 + ATe of 1¾. If there exists a ^-lifting A: K —> K + Ke of
1a: such that Ao = A|k(), then the derivation <5 e Y)zrk{K.K) corresponding to A
induces <?o, i.e., S\Kll = So. We shall show the existence of A. Since K/Ko is a
separable algebraic extension, we have K = K0(9) = Ko[t]/(f(t)). where./(f) is
a separable polynomial, i.e., f'{6) 7^ 0. Let fs"(t) be the polynomial obtained by
applying ^0 to the coefficients of/(/). Let A(0) = 0 - (fs»(6)/f'(e))e. Then it
is readily ascertained by a simple computation that A(/(0)) := /A(I(A(0)) = 0.
where /A° is the polynomial with coefficients in Kq + Ke obtained by applying Ao
to the coefficients of /. Thus, we know that Ao is extended to A: K —> K + Ke.
Consequently, we have 0.Ko/k ®^a K = Q-K/k- Meanwhile, with the notations in (1).
we have Q,g/k ®ff K = QK/k (Lemma 6.17 (2)). We have, therefore.
dim*-,, QKlj/k = dim* QK/k = dim* Q#/A. ®ff K < dimA.(v) Qff/k ®0 k(x).
(In order to prove the last inequality, one may use the following remark: Let M be a

148
11. SCHEMES AND ALGEBRAIC VARIETIES
finitely generated ^-module, and let u\ u, be elements of M such that u\ ur
(iij = itj (modraM)) form a linear independence basis of the A;(x)-module M/mM.
Then M = Y^i -\ &ui by vn"tue of Nakayama's lemma.) Here it is easy to show that
DerA (*„. K0) = DerA (A;[x, x„], K0) =K0d{+--- + K0dn (¾ = ^-) and 3, d„
are linearly independent over Ko. For example. d\ d„ are linearly independent
because a\d\ -\ • • ■ I a„d„ = 0 (a, € Kq) entails (a\d\ + ■ ■ ■ + and„)(x,) = a, = 0.
Hence, n = dim*,, QKo/k- a
Lemma 6.20. With the same situation as in Lemma 6.19. the following assertions
hold:
(1) For a k-rational point x 6 X. Qx/k.x ®tf k — m/m2.
(2) Let X he an algebraic variety defined over k. In the case x is a k-rational
point of X, X is nonsingular at the point x if and only ifQx/k.x '■? a free &x.x-module
of rank n (= dimX).
Proof. (1) It suffices to show that the following homomorphism of ^-modules
d*: \ytxk{&.k) —> Hom^m/m2. A;)
is an isomorphism, where d assigns to 8 e Derk(&.k) an element u : = d (8) such
that u(~z) = 8(z) (z e m, z = z (modru2)). Clearly, d is injective. Conversely, given
u 6 HomA (m/m2. A:), define a homomorphism of A;-modules 8: & —> k by 8(a) = 0
(for each a 6 k) and 5(a) = u(a) (a em, a = a (modm2)). Then 8 is a ^-derivation
of & with values in A;. By the construction, d (8) = u. Hence, d is an isomorphism.
(2) The "only if" part. Since &x.x 1S a regular local ring, we have dim^ m/m2 =
n = dim X. Moreover, the equalities hold in the assertions (1), (2) of Lemma 6.19.
By Nakayama's lemma, we have Q^ = @5\ + ■ ■ ■ +(98,,. Since Clfc/k = &&/k ®@ K,
we have Qx/k — K8] + ••• + K8„. Since dim^ Qx/k = n,8\ 8„ are linearly
independent over K. So {<?i 8„} is a free basis of £l#/k-
The "if" part. By the hypothesis, dim,t m/m2 = n. So m is generated by n
elements by virtue of Nakayama's lemma. Hence. & is a regular local ring. So x is
a nonsingular point of X. □
As a corollary of Lemma 6.20 we obtain the following result.
Theorem 6.21. Let k be an algebraically closed field, and let X be an algebraic
variety of dimension n defined over k. Then X is nonsingular if and only ifQx/k "' a
locally free <0>x-Module of rank n.
Proof. The result follows straightforwardly from Lemma 6.20 if we note that
the set X(k) of Ac-rational points is dense in X and that a quasicoherent ^-Module
3~ is a locally free (f^-Module of rank n if and only if 3~x is a free &x x-Module for
each x e X(k). □
Let k be an algebraically closed field, and let X be an algebraic variety defined
over k. As shown in Lemmas 6.19 and 6.20. Qx/k.x ®e> k — na/m2 and Der^(<f, k) =
(m/m2)* := HomA.(m/m2. k) for a closed point x of X. We call (m/m2)* the Zariski
tangent space of X at the point x. Let A;[e] = k+ke (e2 = 0) and let Homk-„.]%(&■ k[e])
be the set of A;-algebra homomorphisms <p: & —> k[e] such that
(fi(a) (mode) = a (modm) for each a e &.

6. NONSINGULAR ALGEBRAIC VARIETIES
149
As shown in the proof of Lemma 6.19 (2), there is then a one-to-one
correspondence between Homk.aig(&.k[e]) ar>d DerA(if.k). Given <p 6 Homk.d\g(0.k[e]).
the associated morphism of schemes "<p: Specie] —> X maps the unique point
of Specfc[e] to the point x of X and the "tangent vector" e* (= the dual element
of e) of Specie] to a "tangent vector" of X at x. Conversely, any morphism of k-
schemes /i: Spec k[e] —► X, which maps the unique point of Spec k[e] to x, is written
as n = "tp for tp e Homt.aig^.fcte]).
Lemma 6.22. Let k be an algebraically closed field, let X be a nonsingular algebraic
variety of dimension n, let Y be an algebraic subvariety of dimension r in X. and let
/ be the defining (radical) Ideal of Y. Then the following conditions are equivalent
to each other:
(1) Y is a nonsingular algebraic variety.
(2) The sequence of'®y-Modules
o^f/^2^nx/k\Y^aY/k^o
is an exact sequence, and f / f2 is locally a direct summand ofQ.x/k\Y- Namely, there
exists an open covering {Vx}xe\ °f Y sucn mat (f I f2)\v, 's isomorphic (via n) to a
direct summand of£lx/k\vA for each A.
(3) For each y 6 Y(k). the sequence of k-modules
0 - (f/f2)y ® k -> (nX/k\y)y ® k -+ nY/k,. ®k^0
is an exact sequence.
Proof. (2) implies (3). Obvious.
(3) implies (1). For y e Y(k). set A = ffx,v. R = &y.y, J = /y, m = mx,y. and
n = m/J. The exact sequence in (3) coincides with the following exact sequence
0 -► J/mJ => m/m2 -^ n/n2 -+ 0.
Since rj is injective, we have J n m2 = mJ. On the other hand, dimj- m/m2 = n and
dirrn- n/n2 > r by Lemmas 6.19 and 6.20. Let s = dimt J/mJ. Then we can choose
a system of parameters {a\ a„} of m satisfying the following two conditions:
(i) a, £J(\<i<s) and a, (modm2) (1 < i < s) constitute a basis of J/mJ,
(ii) the images of a, (modm2) (s + 1 < i < n) in n/n2 constitute a basis of n/n2.
Applying Nakayama's lemma, we know that J = (ci\ as). Similarly, n =
(a.v+i a„) (a, = a, (mod/)). By Krull's altitude theorem (Lemma 6.8), ht(/) <
.s. Here note that J is a prime ideal. Thus, dim/? > n — s. (Here we make use of
the following result: If A is a localization of a finitely generated fc-algebra domain,
any maximal prime ideal chain has the same length, which is equal to dim A. See
[5] for this result.) Meanwhile, n — s > r = dim/? by Lemma 6.20. Hence, n — s =
r. Therefore, R is a regular local ring, and {as+\ an} is a regular system of
parameters of R. We can alter the above arguments as follows. Since A is a regular
local ring and {a, a„ } is a regular system of parameters, the m-adic completion A
of A is equal to a formal power series ring k[[a\ a„]]. while the n-adic completion
R of R is given as R — A/J A. In fact, since R is a finitely generated /i-module.
R = R®A A = A/J A (Lemma 6.2). Hence. R = k[[as+] a„]] is a formal power
series ring in (n - s) variables. This implies that dim/? = n - s (Lemma 6.11) and
R is a regular local ring (Theorem 6.12).

150
II. SCHEMES AND ALGEBRAIC VARIETIES
(1) implies (2). We employ the same notation as above. We have to show that
the sequence of /^-modules
o -> ///2 A n4/k ®A r -> nR/k -»o
is an exact sequence. For this, it suffices to show that n is injective. Applying ®/?/n
to the above sequence, we obtain an exact sequence
J/mJ -¾ m/m2 -► n/n2 -► 0.
Since Im?/ = /// Dm2.77 is injective if / Dm2 = m/. We shall prove that J Dm2 = ra/.
There exists a regular system of parameters {«1 a„} of A such that [ax+\ «„}
(a, = «,■ (mod/)) is a regular system of parameters of R. We may assume that
a\ as £ /. It is clear that m/ C / n m2. We shall show the opposite inclusion.
Let /' £ /. As an element of ^4 = fc[[ai «„]]. /' is expressed as a formal series in
a\ a„ with coefficients in k. Since R = k[\as+\ «„]] and f (mod J A)) = 0. we
have /(0 0. a,+ , «„) = 0. Hence, if/ e / Dm2, then f'e mJ + mN(N » 0).
Meanwhile. / D m* C m/(7V > 0) by the lemma of Artin-Rees (1.1.31). Hence.
/ e m/. Namely. / n m2 C m/. In view of the arguments employed in the step
showing that (3) implies (1). we show that J = (a\ as). Here Q^ is a free
/4-module with {<i«i da„} as a free basis. Hence. O.A/k ®a R is a free /?-module
with {(/«i g?«„} as a free basis, where da, = daj (modJQA/k). Noting that
///2 is generated by a, (mod/2)(l < i < s) and //(«, (mod/2)) = t/a, (1 < * < s).
we know that ///2 is a direct summand of £lA/k ®A R. D
For an algebraic scheme X over an algebraically closed field k. we call O-x/k
the sheaf of differential forms (the differential sheaf, for short) of X. Suppose X is a
nonsingular algebraic variety. The dual sheaf &~x/k '■= %?t>m&x (&x/k -^x)is called the
tangent sheaf of X. If x is a closed point of X. then &~x/k.x ®ff k — (m/m2)* because
O-x/k is a locally free (f^-Module. where <f = ^,, and m = mi..,, &~x/k.x ®ff k is
therefore identified with the set of tangent vectors of X at the point x. Under the
situation of Lemma 6.22. if the equivalent conditions are satisfied, the coherent <9\-
Modules ///2 and (f'/f2)v := Homgr(f/f2.^Y) are called the conormal sheaf
and the normal sheaf of Y in X. respectively. These are locally free (fy-Modules.
We denote {f jf2Y also by J^y/x- Under the same circumstances, we have the
following exact sequence of (fy-Modules:
0 -+ &Ylk _> g-xjk 1Y _> yrr/x ->0.
The next objective is to define the canonical sheaf a>x/k '■= A" &x/k f°r a
nonsingular algebraic variety X defined over k. For this, we shall recall the notion
of exterior products. Let R be a ring, let F be a free /?-module of rank n, and let
{e\ e„} be a free basis of F. For an integer r(\ < r < «). the r-fold exterior power
/\r F of F is a free /?-module generated by a free basis {etl A • • • A e,- ; 1 < i\,.... ir < n)
such that
eii A • • • A eil A ei/+l A • • • A <?,- = -eh A • • • A eii (, A ei/ A • • • A e,- .
ei] A • • • A £'/, = 0 if fy = ii for some 7 and I (j ^ I).
Therefore, /\' F has rank ("). In particular, if r ~ n. /\" F is a free /?-module of

6. NONSINGULAR ALGEBRAIC VARIETIFS
151
rank 1 generated by e\ A • • • Ac,,. Let {f\ /„} be another free basis of F. Write
'(/. fn) = A'(c{ e„). AeGL(n.R).
Then /i A ■ ■ • A /„ = det(A) • e\ A ■ • ■ A e„. where det(^4) £ R* ={invertible elements
of R).
Now let X be a connected scheme. We shall define the n-fold exterior power
l\" &~ for a quasicoherent. locally free (fy-Module of rank n. By hypothesis, there
exists an affine open covering {C/;.};.6a of X such that 9~\u, — ^tf" for each k e A.
Write t/; = Spec(v4;.) and ^\u, = My where A/; is a free A-,,-module with a free basis
{e\ e'„}. We then set A"(-^"U',) = &v, • < A • ■ • A <. If we write on UA n £/,,.
'« <) = ApSiei <■). ^,;. G GL(«.r(c/, n £/„.<?*)).
then we have
e\ A ■ ■ • A e£ = detC^Jef A • • ■ A e'n.
fMr.= det(AMk)£r(u;nuM.#xy.
and /,.;. = /,.,, • /^; on U-A nt/,,n £/,,. Hence. {A" (-^¼)^ e M 8et patched
together to form an invertible sheaf A" & • The ambiguity arising from the choice
of an open covering {£/;.};.6a and a free basis {e( e'n} disappears if we consider
the isomorphism class of A" &• So & determines an invertible sheaf A" & UP to
isomorphisms. Furthermore, we have an isomorphism of invertible sheaves /\m 'S =
(A" &~) ® (A' ^) if we are given an exact sequence of locally free ^-Modules
where &.12.2? have rank n.m.r (m = n + r). respectively.
Let k be an algebraically closed field, and let X be a nonsingular algebraic variety
of dimension n defined over k. Since Qx/k is a locally free ff^-Module of rank n, we
can define the «-fold exterior power a>x/k :~ A" ^x/k- We call this invertible sheaf
the canonical sheaf of X. We denote it also by ojx- Denote by Kx a Cartier divisor
which gives rise to cox/k ■ The divisor Kx is determined by X uniquely up to linear
equivalence. We call Kx "the" canonical divisor of X. We have ojXjk = &X(KX).
Let x be a closed point of X and let {x\ x„} be a regular system of
parameters of the regular local ring &x.x- If we regard each x, as an element of the
function field k(X). then there exists an open neighborhood U of x such that for
each x; € T{U.^X). If y is a closed point of U and {y\ y„} is a regular system
of parameters of «fXi., then we can express dx, = Yl"j = i aU^yi ^n ^x/k.v where
a,j e @X.y Write «,-,■ = |^. and set
^ X") := det (p.)
>'i yJ \dyjJi<i.i<n
Then we have
(x l . x \
<ixi A • • • A <ix„ = / " ——- dy\ A • • • A <ij„.
\y\ yn)
Write (c/.vi A • • • Adx„)(y) ^- 0 if the value of/(^^) at the point y is nonzero. By
replacing U by a smaller one if necessary, we may assume that {dx\ A- • ■ Adx„){y) ^ 0
for each y e U(k). We then call {x\ x„} a system of parameters at the point x

52
II. SCHEMES AND ALGEBRAIC" VARIETIES
with a coordinate neighborhood U (or simply a system of local coordinates at x). In
sum. if j e U(k). then {xi - x\{y) xn - x„(y)} is a regular system of parameters
of @xA • If jc varies, then X is thus covered by coordinate neighborhoods as defined
above.
Example 1. cov«/k — &{—n - 1).
In fact, let (7b T„) be a system of homogeneous coordinates on P". Then
P" = \Ji=0D+(Ti) mth U, = D+(Ti). We can take {7b/7, (T-/T,)
T„/Tj] as a system of parameters on C/,. Since 7)/7} = Tj/Tq • (7,/70)~l. a simple
computation gives
Therefore coP„/A = <?(-« - 1).
Example 2. Let & be an algebraically closed field, and let V be a nonsingular
projective algebraic variety of dimension 2 (called an algebraic surface). Let C be a
nonsingular complete algebraic curve "on" V, which synonimously means that C lies
on V as an algebraic subvariety. The defining ideal f of C is nothing but the sheaf
&v{-C), ■And///2 =@V{-C)®gv@c. So we obtain coc/k = {coy/k ®<MC)) ®#
<fc by virtue of Lemma 6.22.
Definition 6.23. Let k be an algebraically closed field, let X and Y be
nonsingular algebraic varieties defined over k. and let f: X —> F be a morphism of ^-schemes.
We say that f is a smooth morphism if / satisfies the following two conditions:
(i) f is a flat morphism.
(ii) For each closed point y e Y. the fiber X,, of f is a nonsingular algebraic
variety of dimension r defined over k(y){= k). where r is independent of the choice
of j.
(The condition (i) implies the assertion in condition (ii) that r is independent
of the choice of y (Theorem 4.30 (3)). Conversely, condition (ii) implies condition
(i). though we will not give the proof.)
We shall treat many examples of smooth morphisms in the Part III. We shall
here state the following result.
Lemma 6.24. Let f: X —> Y be a smooth morphism as in Definition 6.23. Then
Qx/y is a locally free @x-Module of rank r, and the following sequence is exact:
0 -> f*QY/k —> &x/k —> Qx/y —> 0.
Proof. Let x be a closed point of X. Set y = f{x), A = &x.x- na = m^..v,
B = (fYA,, and n = mrY. Then A = A/nA = ^.v and m := m/nA = mx,.x- Let
n = A\mX and m = dim Y. Then r = n—m, and X. Y. Xv are nonsingular algebraic
varieties of dimension n.m.r. respectively. Hence.
{f*0.Y/k)x ®k = n/n2. Qx/k.x ® k = m/m2. and QX/y.x ® k = m/m2.
First of all. we shall show that the following sequence
0 -> n/n2 ^ m/m2 -♦ m/m2 -^ 0

6. NONSINGULAR ALGEBRAIC VARIETIES
153
is exact. Choose a regular system of parameters (u\ um) of the regular local
ring (B.n) and a regular system of parameters (u,„n u„) of the regular local
ring (A.m.) so that m, e m and u, = u,- (modn^4) (m + 1 < i < n). Since the sequence
of fc-modules
/ 2 V /2 — i—2 n
n/n —> m/m —> m/m —> 0
is exact (cf Lemma 6.17 (3)). u\ u„ generate the ideal m. Since n = dim^4.
(u\ u„) is a regular system of parameters of the local ring (A.m.). So^isinjective.
Then we can write Q.B/k ®b A ~ Y17=\Aduj. Q.A/k = Y^%\Adu; and Q.a/b =
5Z/=m+i ^ dui (compare the proof of Lemma 6.20). Hence, tp is (locally) injective.
and Qa/b is a free ^-module of rank r. □
Corollary 6.25. With the same hypothesis of Lemma 6.24. we have cox/k =
r(ojY/k)®/\rQ.x/Y.
Proof. Clear from Lemma 6.24. □
Definition 6.26. Let k be an algebraically closed field, let f: X —> Y be a
morphism of algebraic fc-schemes, and let x be a closed point of X. We say that
f is unramified at the point x if Qx/y.k ~ (0). that f is an unramified morphism if
Qx/y = (0)- and that f is an etale morphism if f is a flat and unramified morphism.
Lemma 6.27. With the same notations as in Definition 6.26, the following two
conditions are equivalent to each other:
(1)/ is unramified at a closed point x of X.
(2) tua-.v = my.yffx.x. where y = f(x).
Proof. (1) implies (2). Put A = @x.x- m = mx.x- B = (9Y.y. and n = my,. Since
Qx/y.x ~ Qa/b = (0). we obtain a surjective homomorphism O-B/k ®b A —* O-A/k —► 0-
So applying ®AA/m to this homomorphism, we obtain a surjective homomorphism
n/n2 —> m/m2 —► 0. Hence, m = nA.
(2) implies (1). With the same notations as above, the hypothesis m = nA
implies Q.A/b ®a A/m = (0). Since Q.a/b 's a finitely generated ^-module, we have
aA/B = (o). □
An unramified morphism, e.g., a closed immersion, is not always an etale
morphism. If X and Y are algebraic varieties and if f: X —> Y is a dominant unramified
morphism. we have dimX = dim Y (Problem II.6.4). Here we shall state the
following result.
Theorem 6.28. Let k be an algebraically closed field of characteristic 0, let X and
Y be nonsingular algebraic curves defined over k. and let f : X —> Y be a dominant
morphism. Let k(X) and k(Y) be the function fields of X and Y. respectively, and let
n = [k(X): k(Y)].
(1) For a closed point x ofX and for y = f(x). &x.\ and&y.x are discrete valuation
rings of k(X) and k(Y). respectively, and <9x.x > ^.,- Let t and u be generators of
mx.x und my.,., respectively. Then u = cf with c £ &x v and e > 1. The morphism
f is unramified at x if and only if e = 1. Moreover, we have Qx/y.x — &x.x/(tl'~1)-
(We denote e by e(x) and call it the ramification index of f at the point x.)
(2) For a closed point y ofY. we have^]=i e(x,) = n. where f ~] (y) = {x\ xs}.

154
II. SCHEMES AND ALGhBRAIC VARIETIES
(3) The following sequence is exact:
0 —> / *£ly/k —> &x/k —► &X/Y -^ 0.
H'/iere Supp(QX/y) z.v a finite set and length(Q^y) = Ylxex(e(x) ~ ^)-
(4) deg^/A = « -degQy/A. + length(Qx/y).
Proof. (1) The first assertion is stated in the fourth chapter and Problem II.4.7.
It should be clear that the number e is determined independently of the choice of t
and u. Since Q-x/k.x = ®x.x dt and £ly/k.v = <9Y.X du. the exact sequence
f*QY/k —> &X/k —> &X/Y ~> 0
yields Q.X/y.x = <9x.\dt/@x.x du = &x.\-/(tL'~1)- (Here we use the hypothesis that
the characteristic of k is zero.) Hence. Qx/y.x = (0) if an<i onrv if e = 1-
(2) f is a finite morphism. This is verified as follows. Noting that k(X) is a
finite algebraic extension of k{Y). let Z be the normalization of Y in k{X). and let
g: Z —> Y be the normalization morphism. Then g is a finite morphism. By virtue
of Theorem 4.23, there exists a morphism h: X -»Z such that f' = g -h. Since h is a
birational mapping between normal complete algebraic curves, h is an isomorphism
by Theorem 4.29. Namely, we can identify f with g. from which we know that f is a
finite morphism. Set B = <fy.,.. n = mYv. and Spec(^4) = X xY Spec(B). Then A is a
finitely generated B-module, and f~x{y) corresponds bijectively to Spec(^ ®b B/n)
as the sets. Meanwhile, B is a discrete valuation ring and A is a torsion-free B -module.
So A is a free 5-module (Problem II.3.11). whose rank n = [k{X)\ k(Y)] is equal
to dim<■ A/nA. Since A/nA is an artinian ring, we have A/nA = ]Tj=1 ^4,-.^4,- —
@x.xjn$x.x, by virtue of the Chinese remainder theorem. (Let n^4 = qi n • ■ • n q.s
be the shortest expression by primary ideals. Then m, := y/qj (1 < / < .v) is a
maximal ideal of A. Hence, q, + q; = A if /' ^ j. Then ^4/n^ — 11/^i^/l'
by the Chinese remainder theorem. Since ^4/q, = AmJqiAmi and q,Am/ = nAmi,
we have ^4/n^4 = 11/ = 1 ^<- Here we may consider Ami = ffx.x,-) Since length
{&x.x,/n&x.Xl) = <-'(*,■). we obtain JXi e (•*,■) ~~~ n-
(3) Let x be a closed point of X, and let y = f(x). With the above
notations, (/*Qr/A-).v — @x.xdu and Q^/i-.A- — @x.xdt. Hence, the homomorphism
tp: f*QY/k —> &x/k is injective if we look at it for the stalks at x. So we obtain the
above-stated exact sequence. On the other hand, since O-x/y ®<s, k(X) = Qk(x)/k(Y)
and k(X)/k(Y) is a separable extension, we have Qk(x)/k(Y) = (0)- This entails that
Supp^x/r) is a finite set. It follows from (1) that length {Q.x/y) = J2xex(e(x^ ~ ')•
(4) Refer to Theorem III.7.5. □
Let {X.stf) be a ringed space, and let &."§ be jaf-Modules. We define a cup
product of Cech cohomologies
U: Hp{X.5r) x fii(X.&) -* Hp+''{X.& ®&)
as follows. Firstly, for an open covering it = {£/,-},-e/, define
U: Cp{ix.Sr) x C{!&.&) -> C+l<(U.9T <g» 3?)
by {(7 U r)(/0 ip+lj) = er(i0 ip) ® ?{iP iP+q). where a e C(il,&) and

6. NONSINGULAR ALGEBRAIC VARIHTIKS
155
t e C'iii.S'). Denoting by d the boundary operators of the complexes C'{ii.9).
C*(U,3?), and C'{ix.9 ®9). we have
d{a U t)(z0 V+</+i) = rfo-('o ip+i) ® ^(v+i V+^+i)
+ (-l)M'o /,,) ®dx{ip //,+«,+1).
The cup product of cohomologies is this way induced as
U: Hp {0,9-) x Hi (0,2?) -^ Hp+l< {<&. 9 ® %).
If an open covering D of X is a refinement of U. the following diagram is commutative:
HP{iL.&-)xHi{ik.&) —^ HP+i(U.&-®&)
I I
HP{£),^) x Hi{£).9) —^--> /7^(£>, ^0^)
where the vertical arrows are the homomorphisms induced naturally by the relation
it < £). So the cup product of Cech cohomologies is defined:
U: Hp{X.9-) x H<<{X,2?) -+ Hp+i<{X.9 &&).
If X is an algebraic scheme over a field k and 9 and 9 are coherent <fy-Modules.
then the cup product is a bilinear mapping of fc-vector spaces
U: W{X.9) xHi{X.&) -+ Hp+il{X.9 ®9).
The following theorem is very useful when we treat cohomologies on algebraic
varieties.
Theorem 6.29 (Serre duality theorem). Let k be an algebraically closed field, and
let X be a nonsingular complete algebraic variety of dimension n defined over k. Then
we have the following results:
(1) H"{X.cox/k)=k.
(2) For every invertible sheaf ' 5C on X. the cup product
U: H'(X.S') xH"-''{X.%-y ®cox/k) -+ H"(X,cox/k) = k
gives a perfect pairing for every i (0 < i < n). Namely, we have H"^'{X.2C^^ ®
wx/k) — H'{X,9)y, where V signifies the dual vector space.
The proof is not easy. A proof of the simplest kind is found in [4, Chapter III].
The reader is advised to compare dimH'{X,S?) with dimH"~'{X. Jz?_l ®>cox/k) in
the case where X = P" and jz? = <9x{p) (cf. Lemma 5.12). As a final subject of the
present chapter, we shall present the following result.
Theorem 6.30. Let k be an algebraically closed field, and let X be a complete
algebraic variety defined over k. Let Do be a Cartier divisor on X. let L C |Z)0| be a
linear system, and let ¢/,: X ■ ■ ■ -+ P" (« = dimL) be the rational mapping associated
with L. Then ¢/, is a closed immersion if and only if the following two conditions are
satisfied:
(1) For arbitrary closed points P, Q of X. there exists D e L such that P £ D and
(2) Let M be the k-submodule of H°{X.(f{Do)) associated with L. For an
arbitrary closed point P ofX, choose an open neighborhood U of P so that an isomorphism

156
II. SCHEMES AND ALGEBRAIC VARIETIES
ip: @(D0)\u ^ @u exists. Let MP = {a e M;p{a){P) = 0}. Then the natural
homomorphism of k-modules Tp: Mp —> rtv/m^> is surjective. where mP = mx.p-
Proof. The "only if part. We may identify X with a closed subscheme of P".
Then any element D of L is regarded as the restriction D = X • H onto J of a
hyperplane H of P" (cf the remark after Lemma 5.19). By the hypothesis n = dimL.
there are no hyperplanes H on P" such that H DX. Hence. M = H°(P" ,0(1)). For
a nonzero element a of H°(P" ,&(1)), the hyperplane H(a) defined by a = 0 cuts out
D(a) = X • H{a) on X. The condition (1) is then equivalent to the condition that for
arbitrary closed points P, Q of X, there exists a e H°{Pn.ff{\)) such that P e H{a)
and Q ¢ H{a). It should be clear that the condition (1) holds. Let (7^ T„) be
homogeneous coordinates of P" such that /-" = (1.0, 0). We may identify Mp with
(E/-i "iT,-: («i a„) e fc"}. Let f,- = 7,/7¾. Then {£?=, «,?,■; (a, a„) e k"}
generates the maximal ideal rap p of &p«,p. Hence, p: Mp —> mp/mp is surjective.
The "if part. Let {(7(,,... ,er„} be a fc-basis of the fc-module M associated
with L. For a closed point P of X, the rational mapping <J>L is expressed as^H
(<p(oo)(P) <p(on){P)) up to a projective translation of P" in terms of the open
neighborhood U and the isomorphism tp: <f (£>o)|{/ ^> <9y. For an arbitrary closed
point P of A\ the conclusion (1) implies that ip{o){P) ^ 0 for some a £ M. So
p := ¢/. is defined everywhere. Namely, p is a morphism. By condition (1), p is also
an injection from X(k) to P"(k). We claim that />: X —> P" is a closed immersion. In
order to prove this assertion, it suffices to verify that p*: ^p».,,(/>) —> ^./> is surjective
for an arbitrary closed point P of X. Put A = @x.p.m = rtv>f./>..B = ^.,,(^) and
n = mpn/;(^). Since/) is a proper morphism (Lemma4.19 (5)), p^&x is a coherent ^-
Module (Theorem 4.18). Hence. (p*@x)P(p) is a finitely generated ffv„ (F)-module.
Noting that (p*@x)/,(p) — @x.p- we thus see that A is a finitely generated fi-module via
p*: B —> A. Condition (2) then implies that m = p*(n)A + m2. Hence, m = p*{n)A
by Nakayama's lemma. So A = p*(B) + m = p*{B) + p*(n)A. Applying again
Nakayama's lemma to a 5-module A/p*(B). we obtain A = p*{B). □
II.6. Problems
1. Let R[x\ x„] be a polynomial ring in n variables over R. Show that
£2fl[Yl Xiiyn is a free R[x\ x„]-module with a free basis
{dx\ dx„}. (Compare the proof of Lemma 6.19 (2).)
2. Let X be an algebraic scheme defined over a field k. Show that if Clx/k ~ (0)-
then dimX = 0. (First, show that we have only to give a proof in the case where
X is irreducible and reduced. Second, show that Q-K/k = (0) if K = k(X).
Finally, show that tr. degA. K = 0.)
3. Let k be an algebraically closed field. Let V be a hypersurface in P"k defined by
F(T(, T„) = 0. Show that the singular locus Sing(K) of V is defined by
dT0 dT„
4. Let k be an algebraically closed field, let X and Y be algebraic varieties defined
over k. and let f:X—> Y be a dominant unramified morphism. Show that
dimX = dim Y. (Make use of problem 2 above.)

Part III
Algebraic Surfaces

CHAPTER 7
Algebraic Curves
Throughout the present part, k denotes an algebraically closed field of
characteristic p > 0. and all algebraic varieties are assumed to be defined over k unless
otherwise mentioned. We call k the ground field.
Let X be an algebraic scheme proper over k, and let !F be a coherent $x-
Module. Then H'(X .9") is a finite-dimensional ^-vector space. We set h'(X.9~) =
dimH'(X9-). If « = dimA-. then ti{X.9) = 0 for each i > n (problem 1.2.8). We
denote by x{X-&~) tne alternating sum ^"^0(-l)'/;'(X.^) and call it the Euler-
Poincare characteristic of !F. All points we treat below are closed points unless
otherwise specified.
Lemma 7.1. (1) If 0 —> 5F\ —> &~j —> 9~-$ —> 0 is an exact sequence of coherent
(fx-Modules, then x.{X.9i) = x{X.9T\) + x{X.9r3).
(2) 7/"dimSupp(^) < Qfor a coherent @x-Module 3~, then we have x{X.3~) =
h\X.&) = length^).
Proof. (1) It follows readily from the cohomology exact sequence (compare Part
I, Chapter 2).
(2) This follows from the remark that h'(X.9~) = (0) for each i > 0 and
/,0(X.,Sr) = £.esupp(^)dim^v. n
Let C be a nonsingular complete algebraic curve. Since each local ring of C
has dimension 1. it is regular if and only if it is normal. The Weil divisor group
Div(C) of C is a free abelian group generated by the set C(k) of closed points. For
an element D = ^, «,-P,- of Div(C). we denote ^. n,- by degZ) and call it the degree
of D. By Lemma II.5.14, D defines the corresponding invertible sheaf ff(D). D is
linearly equivalent to an effective divisor if and only if H°(C.&(D)) ^0. If D itself
is an effective divisor, then there exists an element a ^ 0 of H°{C.&(D)) such that
D = D(a). For each point P of C. there exist an open neighborhood U of P and
fu e r(I/.<?c) such that D is defined by fv = 0 on U. Then &{D)\V = (fu(fu)~i
and a\u = fu • (/[/)"'• Since H°(C.ff{D)) = Honvr{<fC-@(D)). a corresponds
to an (fc-homomorphism a: <9C —> <9{D), and a is given as <r|c/(l) = /<y • (/[/)"'•
We denote this ffc-homom°rpmsm «5 by the same symbol a. It is readily shown that
a is injective, (f{D)/a((fc) = @c/@{-D) = <9D, and degZ) = length@CI@{-D).
Applying Lemma 7.1 to the exact sequence
0 -► &c "^ ^U>) -> &{D)/a(&c) -^ 0.
we obtain /(C. <f (£>)) = x(C.<fc) + degZ). If Z> is not effective, write D as D =
D\ - Z)2 with D\ > 0 and D2 > 0. Applying <E>^c^(Z>i) to the exact sequence
0 _► <?(-£>2) -+ @c ~+ @d2 -+ 0.
159

160
III. ALGEBRAIC SURFACES
we obtain the exact sequence
0-><?(£>) -^ODi) -^<9Dl ->0.
where <f£>, ®&c <f(D\) = (¾.. because Supp(<f£>,) is a finite set. Hence. Lemma 7.1
yields
/(C.^(/)))=/(C.<?(/),))-deg/)2
= x(C.&c)+degD\ -deg£>2
= ^(C.(?r)+deg£>.
In the equality
yXC.&c) = /»°(C.f?>c)-/iI(C.^c) = \-h\C.ffc).
we have hl{C.ffc) = h°(C.coC/k) by virtue of the Serre duality theorem (Theorem
II.6.29). We write g(C) : = h[(C.@c) and call g(C) the genws of C. Summarizing
the above observations, we obtain the following theorem.
Theorem 7.2 (Riemann-Roch Theorem). Let C be a nonsingular complete curve
of genus g. and let D be a Weil divisor on C. Then we have:
(l)X(C.&(D))=degD + \-g.
(2) h'(C.0(D)) = hl-'{C.^(Kc - D)). i = 0.1.
(3) h\C.@{D)) = 0 provided degD < 0.
Consider next a singular complete algebraic curve C. Let C be the normalization
of C and let v: C —> C be the normalization morphism. Then v is a finite morphism,
so it is an affine morphism. Hence, we have R' vj? = (0) for each i > 0 for a quasi-
coherent ^-Module 'S (compare (11.3.23)). Therefore. W{C.9) = //'(C. v»2?) for
each i > 0 by (1.2.27) and (1.2.28). We can verify this assertion by means of Cech
cohomologies and (II.5.11). without using the Leray spectral sequence.
Let 2f be an invertible sheaf on C. Then v*2f is an invertible sheaf on C. So
there exists a Weil divisor D on C such that v*S? = &(D). We define degjif = degD.
We shall generalize the Riemann-Roch theorem on C. We define the arithmetic genus
(or virtual genus) of C by p(C) := hx(C.@c). We denote hl(C.^) by g(C) and
call it the geometric genus of C. In order to elucidate the relation between p(C) and
g(C). consider an exact sequence of ^c-Modules
0 -» &c -> v,^ -► m -► 0.
where we set 91 : = v*<f^/<fc- Noting that dimSupp^ < 0. we obtain
x(C.#d)=X(C.0c) + length^
by taking the Euler-Poincare characteristics. Since h°(C.^) = h°(C.&c) = 1- we
have p(C) -g{C) = lengths. So we have the equivalence: lengths = 0 <=> 91 =
(0) o- C = C <^> C is nonsingular.
Theorem 7.3. Le? C a«(/ v be the same as above. Then the following assertions
hold:
(1) If SF is a quasicoherent &c-Module. we have an isomorphism &~ <g> v*@c =
\uv*y.

7. ALGEBRAIC CURVES
161
(2) For an invertible sheaf 3f on C. we have
x(C.2') = deg2' + l-p(C).
Proof. (1) Define the natural homomorphism
a: 9~ ® v„^~ —> v^gr
by
rit/.jjgrft/j/^rfr'lc/).^). f ®a^+af.
Let U = Spec(/1) be an affine open set of C. let v~l{U) = Spec(/l). and let M =
r(C/..T). Then (.^<g> v*^~)|f/ = {M®AA)~~ and (v*v*F)|L; = (M®AA)~. Hence,
qJu is an isomorphism. So a is an isomorphism.
(2) We have an exact sequence
o -► se -+ % ® v*&£ -+ m <g> se -+ o.
where m <g> ^ = m because dimSupp^ < 0. It follows that /(CS* <g> v*^~) =
/(C.v*.S*) = /(C..S*) + length(^). By Theorem 7.2, X{C. v*S?) = 1 - g{C) +
degi?. Since p{C) - g(C) = length^) as remarked above, we have the equality as
stated above. □
Let C be a nonsingular complete algebraic curve of genus g. For a divisor D on
C. \D\ denotes the complete linear system determined by D. For points P, (1 < i < n)
and positive integers m, (1 < / < «). we denote by \D — Yl"--\ miPi\ + Y11=\ m>^i
(or simply, |D| - £"=1 m,P,) be the linear system {D' G \D\\D' - £"=1 m,P, > 0}.
So, Ym = \ m'P' *s contained in the fixed part of \D - £)"=1 MjPj\ + J21=i miP>-
Theorem 7.4. {\) degKc = 2g - 2.
(2) We have the equivalence: g — 1 iff Kc ~ 0 iffcoc/k — @c-
(3) Lef D be a divisor of degree d. If d > 2g - 2 (or if d = 2g ~2 and D •* Kc).
then dim|D| = d - g.
(A) If d >2g -\. then \D\ has no base points.
(5) Ifd > 2g. then D is a very ample divisor. Namely. ¢^ : C -+ FN(N = dim \D\)
is a closed immersion.
(6) A nonsingular complete algebraic curve is a projective algebraic variety.
(7) IfS is a finite set of closed points of C. then C — S is an affine algebraic curve.
Proof. (1) By the Riemann-Roch theorem. x(Ca>c/k) = deg#r + I - g. and
by the Serre duality theorem. x(Ccoc/k) — -%{C-@c) = g - 1- It then follows that
degKc =2g-2.
(2) Since h°{C.wC/k) = g. \KC\ ^ 0 provided g > 1. Suppose g = 1. Let a be
a nonzero element of H°{C. coc/k). let a: @c -+ coc/k be the corresponding injective
homomorphism. and let D = D(a) which is an effective divisor. We have an exact
sequence
0 -+ ®c -^ 0>C/k -+@D->§-
where length^ = degZ> = degA^c = 0. So &D = (0) and coc/k — @c- Conversely,
if a>cjk — &c- we clearly have 2g — 2 — 0.
(3) By the Riemann-Roch theorem, h°(C.&(D)) = h0{C.^c(K -£>)) + 1 -g +
d. Since deg(A:c - D) < 0 by the hypothesis degZ> > 2g - 2. we have h°(C.@c(K -
D)) = 0. Hence, dim|D| = h°(C.0{D)) - 1 = d - g. Suppose ¢/ = 2g - 2 and

162
III. ALGEBRAIC SURFACES
h°(C. 0(Kc — D)) > 0. By the same reasoning as in (2) above involving the injection
a: @c -+@{KC -D) and an effective divisor D{a). wehave(f(A:r -D) =&C- Hence.
we know that h°(C.&(Kc - D)) = 0 provided Kc * D.
(4) For an arbitrary point P, deg(Z) — P) > 2g — 2. So dim \D\ — d - g and
dim \D - P\ — d — g - I. This implies that P is not a base point of \D\. In fact,
the linear subsystem \D - P\ + P has the same dimension as \D - P\.
(5) For arbitrary points P.Q of C. dim\D - P - Q\ = d - g-2 <d -g-I =
dim \D - P\. Hence, there exists a divisor D' e \D\ such that P e D' and Q ¢ D'. Let
M c H°{C.&{D)) be the £-submodule associated with the linear subsystem \D\- P.
and let U be an open neighborhood of P such that an isomorphism <p: & (D) | L> ^+ &u
exists. Suppose ip(o) e mj> whenever er e H°(C.0(D)) satisfies ip(a) e m/>. Then
|D - 2/>| + IP = |£> - P| + P and this is a contradiction. So by (II.6.30). <D|B| is a
closed immersion.
(6) For an arbitrary point P off. (2g + l)P is a very ample divisor. Hence.
C is isomorphic to a closed subscheme of P?+1 via ^\(2H+i)p\- So C is a projective
algebraic variety.
(7) Write S = {P\ P,.}. Choose an integer m so that mr > 2g. Then
D : = m{P\ + ■ ■ ■ + Pr) is a very ample divisor. So. <D|B|: C —> P^ (N = mr - g) is
a closed immersion, and D — C ■ H for a hyperplane H ofPN. Hence, C - S is a
closed subscheme of A^ := P^ — H. So C — £ is an affine algebraic curve. □
Theorem 7.5 (Formula of Riemann-Hurwitz). With the hypotheses of Theorem
II.6.28, we have the following equality:
2g(X) - 2 = n(2g(Y)-2) + £(«?(*) - 1).
vex
Proof. By (II.6.28 (3)). we have x{X.coX/k) = xix■ f*mY/k) + lengthfi^f.
Since x(X.cox/k) = g[X) - 1 and x(X.f*coY/k) = degf*coY/k + l-g(X). we have
only to show that deg/*coy^. = n(2g(Y) - 2). For this, it suffices to show that
d<zgf*@{D) = n • degD for an effective divisor D on Y. (For an arbitrary divisor
D. write D = Dt - D2 with Dx > 0 and D2 > 0. Since &{D) = @{DX) ®&{D2)^
and f*0(D) = f*&{Dx) <g> {f*^{D2))~x. it suffices to show that the equation
deg/*<f (D) = « • degD only for an effective divisor D.) For Z) > 0. look at an
exact sequence
(*) 0->^r-^r(-D)->^d->0.
As in the proof of (II.6.28 (2)). since f: X —> 7 is a flat morphism. it follows that
length(/*^- ®#r £(>>)) = n f°r eacn J 6 Y(k). The exact sequence (*) yields an
exact sequence
0-+tfx -+ f*<f(D) -+ f*@D -► 0.
If we decompose &D into a composition series and make use of the above remark,
we know that length/*^, = n ■ (length^). Hence. degf*@(D) = n • degD. □
We can elucidate the structures of algebraic curves with small genus.
Theorem 7.6. Let C be a nonsingular complete algebraic curve of genus g. Then
the following assertions hold:
(1) The following conditions are equivalent to each other:
(i) g = 0.

7. ALGEBRAIC CURVES
163
(ii) C ^P1.
(iii) k(C) is a rational function field over k.
(2) Suppose the characteristic of k is different from 2 and 3. If g = 1, C is
isomorphic to a curve on the projective plane defined by the following cubic
equation F = 0:
F = X0Xl - (X,3 + aXlX\ + bX^). a.b G k.4a3 + 27b2 f- 0.
Proof. (1) (i) implies (ii). Let P be a point of C. Then ¢^: C —> P1 is a
closed immersion by virtue of Theorem 7.4 (5). <&|/>| is an isomorphism.
(ii) implies (i). If C = P1. then a>c//t — <^(-2) (Example 1 in the Part II, Chapter
6). Hence, degcoC/k = -2. So g — 0.
(ii) implies (iii). Obvious.
(iii) implies (i). By hypothesis, C is birational to P1. So C = P1.
(2) Let P be an arbitrary point of C. To simplify the notation, we denote
H°{C,ff(mP)) by H°{mP) and its dimension by h°{mP). If m > 0, then /j°(mP) = m
by the Riemann-Roch theorem. We may also assume H°(mP) ck(C) (compare Part
II, Chapter 5). Let {1,£} be a jfc-basis of H°(2P), where £ G fc(C). Likewise, let
{l,4,n} be a A:-basis of #°(3P). (2P + (¢) > 0 implies 3P + (¢) > 0.) Since
<p : = 0|3P|: C —> P2 is a closed immersion, we may assume that £ = <p*(Xi/Xo)
and ?7 = ^{Xj/Xo), where (Ao,Zi,Z2) are homogeneous coordinates of P2. So
&(C) = k(£,.n). As \,£,£2,n G H°{4P), suppose these are linearly dependent over
k, i.e., a + bi + c£2 + dn = 0 (a, b,c,d G fc). Then rf = 0. (If rf ^ 0, then n G Jk(^)
and k(C) = k(£,,n) = k[£). So g = 0 which is a contradiction.) So the relation
a + b£, + c£2 = 0 holds infc(C). So £ = a £ k which is also a contradiction. We
therefore know that {1. £,, £2. »7} is a jfc-basis of H°(4P). Consider next H°(5P). We
have \,S,.S,2.n.S,n G H°(5P). By the same reasoning as above, {1. £.^.77.^77} is a
fc-basis of H°(5P). Consider H°(6P). We have seven elements X.i^.^.^.n.^n.n2
in H°(6P). Hence, there is a nontrivial relation with coefficients in k,
(*) a?72 + b£,n + en = d£ + e£,2 + /£ + g.
If a = 0, then k(C) = k{£,), a contradiction. So we may assume a = 1. By replacing
n by n' = n + {bS, + c)/2, we may assume b = c = 0 as well. If d — 0. then (*) is a
quadratic equation in £ and »7, and fc(C) becomes a rational function field (Problem
III.7.1). Hence, d f 0. Replace £ by £' = (d1/3)^ to assume ¢/ = 1. Finally,
a change of variable £ i-> £' = £ + (e/3) allows us to assume ^ = 0. Namely,
by a linear transformation in £ and »7 (more precisely, an affine transformation),
the relation (*) is changed to an equation rj2 = £3 + a£, + b. By expressing this
equation in homogeneous coordinates, we obtain the equation F = 0 as given above.
Furthermore, C is nonsingular if and only if 4a3 + 27b2 7^ 0 (compare Problem
II.6.3). □
A nonsingular complete algebraic curve C is called a rational curve (resp., an
elliptic curve) if g = 0 (resp., g — 1).
III.7. Problems
1. Suppose the characteristic of k is different from 2.
(1) Show that a conic C on P2 (i.e., a curve defined by F — 0, where F is a
homogeneous polynomial of degree 2) is isomorphic to one of the following:
(i) X0X, = X2,

164
III. ALGEBRAIC SURFACES
(ii) X0X{ - 0.
(ill) Z02 = 0.
2. Show that C is isomorphic to P1 if it is irreducible and reduced.
3. Find the standard forms of quadratic hypersurfaces of P3.

CHAPTER 8
Intersection Theory on Algebraic Surfaces
In what follows, algebraic surfaces signify nonsingular projective algebraic
varieties of dimension 2 unless otherwise mentioned. Let V be an algebraic surface. For
a quasicoherent ^/-Module &. we employ the abbreviations H'{9~) = H'(V.9~).
W{&) = tiiV.F), and/(^) = x{V,&).
Definition 8.1. For invertible sheaves 5C\ and .2¾ on V. we set
(^, • j?2) = x&v) -x(^r') -/(^1)+/(^1 ®^:{)
and call it the intersection number of .2¾ and .2¾.
By definition, (.2¾ • .2¾) is an integer. The following result is important.
Theorem 8.2. (1) ( • ) is an integer-valued binary form on Pic(F). Namely, we
have:
(i) (.2¾ • .2¾) depends only on the isomorphism classes of 5C\ and .2¾.
(ii) (J?,-^2) = (^2-^1).
(iii) {# ® S£{ • j?2) = (s?i ■ sr2) + {sr[ • se2).
(iv) (^,-^^2) = -(^1-^2).
(v) (#v.&2) = o.
(2) Suppose //0½) ^ (0) for i = 1.2. Write 2!-, = 0V{D{) with Di > 0. If
D\ and D2 have no irreducible components in common (no common components, for
short), then we have
CS?, -&2) = ^dimk0v,x/(fx,gx).
xev
where fx = 0 and gx = 0 are local defining equations at x of D\ and D2, respectively,
and dim^ stands for the dimension of a k-vector space.
Proof. (1) (i), (ii), (v) are clear from the definition. We shall verify (iii). Let C
be an irreducible algebraic curve (lying) on V. Consider the case where .2¾ = &V(C).
From the exact sequence
0 -> 0V(-C) -► 0v -> 0C -> 0,
we obtain an exact sequence
0^^71 ®@v(-C)-^#-x -+{S?i®0c)~X ->0.
By Theorem 7.3, we have
(*v(c)-&)={x(*v)-x(M-c))}
-{x(5f-l)-x(Sf-l®0v(-C))}
=*(*c) -/((-¾ ®0c)'{) = deg(^, ®@c)-
165

166
III. ALGEBRAIC SURFACES
Meanwhile, if v: C —> C is the normalization morphism. we have
deg((^, ®y2)®#f) = deg(v*(^, ®&c) ®o- v*(22 ®#c))
= deg v*(J?i <g> ^c) + deg v*(5¾ ® <?c)
= deg(^, <g> ^c) + deg(^2 <g> &c).
These observations imply that (&y(C) • i?) is linear with respect to SC.
Next we claim that given an effective divisor D = £2; a> Q {a,- > 0. C, is
irreducible), (&y(D) • 3?) is linear with respect to 21. The proof is by induction
on Ylj ai- By definition, we have that
(j?i • sei) + {&[ ■ se2) - {&\ ® se[ • se2)
= x(&v)-x&l-l)-x(2"l-])-x(^)
+ x((%'\®2'2)-l)+x(W®2'2)-])+x((2'\®2'irl)
-x{{&\®&{®&2)-1).
The right side of this equality is symmetric in S\,3"x, and S£2. Since the left side = 0
if .¾ = @v{C) (C is an irreducible divisor), the left side = 0 also if %[ = @y{C).
Hence. (2i ® &y(C) ■ 2t2) = (-S?i ■ 2!2) + (@y(C) ■ 22). If we write D = /J), + C
with D\ > 0 and C an irreducible divisor. (@y{D\) - 5C2) is linear with respect to 3!2
by the induction hypothesis. So {&V(D) ■ 5C2) = {@v{D\) • 22) + (0y(C) • 22) is
also linear with respect to 5f2.
We shall now show that {5C\ • .2¾) is linear with respect to 5C2 for an arbitrary
5C\. As V is viewed as a closed subscheme of the projective space PN. we let $y{\) =
@vn(X)®@v and 2\(m) = 2\ ®&v{m). If m » 0. then H°(V^](m)) f. (0). (If
we write 5C\ = @v{D\ - D2){D\ > 0. D2 > 0), then there exists a hypersurface F of
degree m0 such that F ^> V and F 2 U^red. where (D2)red = Xw G ^ D2 = Xw afG
and {D2)rt&7 identified with Supp(Z>2)red, is viewed as a closed subscheme of PM. If
we take m = am0(a » 0), we have a(F • V) > D2, where F • V is the effective
divisor on F obtained by restricting the local defining equations off onto V. Since
@v(aF ■ V) = &v{m), we have 2\(m) = &y{Dx + {aF ■ V -D2)).) Since the left side
of the equality (*) is zero if 5C2 = &y(m), by making use of the symmetry in S?\,5f[,
and 2f2, we know that the left side of the equality (*) = 0 even if 5C[ = &vim). So
we have
(J?, • 3f2) = (J?i (m) • J2z) - {0v(m) • S?2),
where the right two terms are shown to be linear with respect to 5C2. Hence, (Jzfj • 5C2)
is linear with respect to 2f2. By virtue of (ii), {S?\ • Sf2) is linear with respect to 5C\
as well, (iv) follows from (iii) and (v).
(2) Look at the following natural sequence of <f>-Modules
0^K$y{-Di -D2) -> ®v(-Dy)@@v(-D2) ^>&v
-► @D\ <8> ^/), -► 0.
For a point x of V, set A = @v.x- f ~ fx, and g = gx, where fx and gx are the
same as in the statement. Restricting the above sequence to the stalks over x, we
obtain the following sequence of ^-modules:
0^fgA^fA®gA±A± A/(f,g) -+ 0,

8. INTERSECTION THEORY
167
where a and/? are defined by a (fgu) = fgu®(—fgu) and fl(fv(Bgw) = fv+gw.
and 7 is the residue homomorphism. We shall show that this sequence of A -modules
is an exact sequence. If at least one of/ and g is an invertible element, the exactness
is easily verified. Suppose f.g e m := mj.A. Then ht((/.g)) > 1. (In fact, suppose
there exists a minimal prime divisor p of (f.g) with htp = 1. Since A is a UFD
(cf (11.5.16)). we can express p = (h). Then h\f and h\g. and the irreducible
component, which is defined by h = 0 near x. is a common component of D\ and
D2. This contradicts the hypothesis. Hence. ht(/.g)) > 1.) Since dim.4 = 2. we
have (/. g) D mM for some N > 0. Noting that / and g have no common divisors
in A. we shall verify the following two assertions:
(i) For u. v e A. if fu + gv = 0, then for some w G A. u = gw. v = —fw.
(ii) For w G A. if /gio © (-fgw) = 0. then u; = 0.
Since A is an integral domain, the assertion (ii) is clear. We shall verify assertion
(i). Since fu = —gv and f.g are mutually prime, we have f\v. Write v = —fw
(w G A). Then u = gw.
By virtue of the above two assertions, the above sequences of/1-modules and &v
Modules are exact sequences. By taking the Euler-Poincare characteristics of terms
of the sequence of ^/-Modules, we have
/(^, ®&d2) =X&v) -x(@v(-Dx))-X(@v(-D2))
+ X(0y(-Dl-D2))
= (@V(D\)-@V(D2)).
Since dimSupp(<fB| ®@D,) < 0, we obtain
*Wd, ® 0d2) = length^, <g> <?D2) = ^ dirn^ /S>Vx/(fx.gx). U
.vSK
For .2", = <fy(Z>,) and ^2=^(-^2). we define (Di-D2) := (¾ --¾) and call it
the intersection number of D\ and Z>2- We also employ the abbreviations (2C1) : = (i? •
S) and (X>2) = (D)2 : = (D • £>). By Theorem 8.2 (1). we have (Dx ■ D2) = (D[ ■ D2)
if D\ ~ D[. With the notation of Theorem 8.2 (2), if D\ and D2 are effective divisors
without common components, then we define i(D\.D2;x) = dim^ @v.x/(fx-,gx) and
call it the local intersection multiplicity of D\.D2 at a point x. Below, we shall apply
intersection theory to concrete examples.
Theorem 8.3 (Theorem of Bezout). Let C and D be (not necessarily irreducible)
algebraic curves on P2 which have no common irreducible components. Then the
following equality holds:
(degC)-(degD) = (C • D) = ^ i(C.D;x).
.y6P2
Proof. We shall first show that (&(1) • <9(\)) = 1. Let (X0.XX.X2) be the
homogeneous coordinates. Let /o-1\ be lines defined by Xo = 0, X\ — 0, respectively.
Since &(\) = &(h) ^ @(lx). we have
(&(\).&(\)) = (/o-/,) = &mk@Pl(u.v) = 1.
where P = (0.0.1) = /0 n /]. u = X0/X2. and v = X\/X2. We choose a line / on
P2. If an algebraic curve C is denned by a homogeneous equation of degree d, then

168
III. ALGEBRAIC SURFACES
C ~ dl (Example (11.5.17.1)). Hence. degC := d = (C •/). So if e = deg£>. we
have
(C • D) = (¢// ■ e/) = de = (deg C) • (degD).
The stated equality follows from Theorem 8.2. □
Definition 8.4. Let C and D be disjoint irreducible (and reduced) algebraic
curves (lying) on an algebraic surface V. Suppose C and D meet at a point x. Let
fx — 0 and gx = 0 be local defining equations of C and D at x. respectively. If
(fx-gx) = m.v (which is equivalent to i(C.D;x) = 1), then we say that C and D
cross normally at x. Then the point x is a nonsingular point of both algebraic curves
C and D. If C and Z) cross normally at every point of intersection, we say that C
and D cross normally.
Corollary 8.5. Suppose two distinct irreducible algebraic curves C and D cross
normally on P2. Then #(C DD) = (degC) • (degD). where #{C DD) is the number
of intersection points of C and D.
Proof. Apply the equality in Theorem 8.3. □
Lemma 8.6. (1) Let V be a projective algebraic variety, and let 2 be an invertible
sheaf on V. For an irreducible algebraic curve C on V. denote by 2\c the restriction
of 2 onto C. If 21 is an ample invertible sheaf then we have deg(Jzf|c) > 0.
(2) Let V be an algebraic surface, and let 2 be an ample invertible sheaf. Then
(22) > 0.
Proof. (1) 2®n\c = {2\c)®n, and if n > 0, deg(^|c)®" > 0 is equivalent
to deg(^|c) > 0. So we may assume that 2 is very ample. Then the natural
mapping ¢^: V —> FN (N + 1 = h°(V.2)) associated with 2 is a closed
immersion. If we choose a &-basis {<ro— .a^} of H°(V.2). then ¢^ is given by
P i-> [ao{P) , ctn{P))- The restriction ¢^\c of ¢^ onto C is the rational mapping
associated with the submodule lm{H°{V.2) -> H°{C.2\C)) of H°{C,2\C). (In
fact, <b#{C) is contained in a linear subspace fl/f (t) of PM, where x ranges over
nonzero elements of H°{V,Sc ® 2), Sc being the defining ideal sheaf of C in V.
and H(x) is a hyperplane of P^ defined by x = 0.) In particular, $^|c is obtained as
a composite of ¢(^^) and the projection from a projective space to a linear subspace.
Clearly, the complete linear system associated with 2\c has no base points. So ¢(^^)
is a morphism. Since O^lc is a closed immersion, so is ¢(^^) (Problem II.5.6).
Hence, 2\c is a very ample invertible sheaf on C. It suffices, therefore, to show that
degi? > 0 when V = C. Now let v: C -+ C be the normalization morphism. Then
there exists exact sequences
0 -+ ®c -> i^f -* -^ -* °'
and
0->2 -> v*v*2 ->3Z ->0
(compare the proofs of Theorems 7.2 and 7.3). We then obtain an injection H°(C,2)
<-* H°{C, v*2). A nonzero element a of H°{C. v*2) defines an effective divisor
D(^ 0) such that v*2 = &^{D). Hence, deg^ = degD > 0.
(2) Take a positive integer n so that 2®n is very ample. Then (22) —
((2®n)2)/n2. Hence, we may assume that 2 is very ample. Then there exists an

8. INTERSECTION THEORY
169
effective divisor D ~ Yl'i=iaiC> ia< > 0. Q: irreducible component) such that
Sf = @v{D). Then we have
r r
= ^a^S? - 0V<,Q)) = ^fl.-deg^ir,).
/=i /=i
and deg(^|c,) > 0 by virtue of (1) above. □
The converse of Lemma 8.6 holds when V is an algebraic surface. We shall state
only the following result without giving a proof.
Theorem 8.7 (Nakai's criterion of ampleness). Let 2C be an invertible sheaf on an
algebraic surface V. For 5C to be ample it is necessary and sufficient that the following
two conditions are satisfied:
(1) For every irreducible algebraic curve C on V. deg(jz?|c) > 0.
(2) (J?2) > 0.
Let V be an algebraic surface, and let P be a closed point. We define a monoidal
transformation (called also a blowing-up) a: V —> V with center at P as follows. Let
(w. v) be a system of local parameters at P. and let U be its coordinate neighborhood
(compare Part II. Chapter 6). Furthermore, changing U by a smaller one. we assume
that {P} = {x e U\ u(x) = v(x) = 0}. Define a closed subset Tv of V x P1 by
?u = {(£?.(ao-ai));2 6 C/.(a0.a,) €Fl. v{Q)a0 = u(Q)a{}
and the natural projection a: Tu —> U by (Q. (qo.qi)) i-> g. Then <7~'(£/ - {P}) ^
C/ - {P} and <t-'(P) = P1. So we patch F - {P} and T^ together along the
open sets U - {P} and rV - <?~}(P) = (J~l(U - {P}) to obtain a nonsingular
algebraic surface V and a birational morphism a: V —> F. Shrinking C/ further
to a smaller open neighborhood of P. we may assume that U is an afHne open set.
Write U = Spec(A), and let I be the defining ideal of the point P. Then I = (w. v).
We shall show that a 1{U) = Proj(0[>o/'). where ®,->0/' is a graded /1-algebra
such that 1° = A and /'' • V C //+>.-In fact. Proj(0^>o/'') = D+(u) U D+(v),
D+{u) = Spec(/l[v/w]) and D+{v) = Spec(^[w/u]). Let To. Ti be indeterminates.
Then there exists a surjective homomorphism ip: A[T0. T\\ —> 0/>o/' such that
ip(T0) — u and ip(T{) = v. and we can identify Proj(0/>o /') with a closed subscheme
V+{uTx - vT0) of ProjU[T0. T,]) = [/xP1 Hence. ct-»(C/) = Proj(0,>o7').
Here V is a projective algebraic variety over k. though we omit the proof. We write
E = a~l(P) and call E the exceptional curve (of the first kind).
Let C be an irreducible algebraic curve through P. and let f = 0 be a local
defining equation of C at P. Let ^(./. be the m-adic completion of &v.p- Then
&v.p = fc[[w.u]]. As an element of @v.p- f K written as a formal power series in u.v
/ = /,+/,+1+----
where /,- is the /th homogeneous part and fM ^ 0. We denote,« by ju(C; P) and call
it the multiplicity of C at P (compare Lemma 8.8). P is a nonsingular point of C
if and only if p(C;P) = 1. (Prove this assertion by making use of (11.6.20).) Now
choose an open covering it = {C/;};eA of V so that C is defined by /;. = 0 on £/;,
where /;. G T{U;.,@V). Then a~l(!d) = {cr~'(^;.};.eA is an open covering of V. The

170
111. ALGEBRAIC SURFACES
effective divisor on a ' ([/;) defined by a* (/;) = 0 with X ranging over A are patched
together to form an effective divisor a*(C) on V. We call o*(C) the total transform
of C. As ff-'(C - {P}) = C - {P}, the closure of a~x{C - {P}) on K' is an
irreducible algebraic curve, which we denote by a'{C) and call the proper transform
of C. Even if an irreducible algebraic curve C does not pass through P. we can define
a*{C) and rr'(C), though a*(C) = a'(C) = C. If D = J2iaici is a divisor on
V. define er*,D. a'D by er*,D — ^, a,(7*C/. a'D = J2/ ci/a'Ci and call them the total
transform, the proper transform of D, respectively. Clearly. o*@v{D) = @y(o*D).
We shall prove the following result.
Lemma 8.8. (1) (E2) = -1.
(2) Let C be an irreducible algebraic curve through P, and let ju be the multiplicity
of C at P. Then we have
a*C =juE + a'C. (a'C-E)=ju. (a*C-E) = 0.
Proof. (1) Let jVC/ v be the normal sheaf of C on V. We shall show that jVCj v
is then an invertible sheaf on C and (C2) = degyfc/i • For this, consider an exact
sequence
0 -► &y(-C) -► @v ~+ @C ~+ 0.
where @V{-C) is equal to the defining ideal sheaf J"c of C. Taking the tensor
product of each term of the exact sequence with &V(—C). we have an exact sequence
0 _+ @V(-IC) -> ^k(-C) -* <M-C)|r -^ °
from which &V{-C)\C = Scl<?c■ So JVC/v is an invertible sheaf on C and jVCj,,- =
<MC)|C. Hence, (C2) = deg^K(C)|c = deg^c/,-.
We shall determine JVe/v- Let (t0. t\) be a system of homogeneous coordinates
on £(= P1). For a point g of E, if ?0 ^= 0 at Q, set w = t]/t0. Then v = uw
by the definition of IV, and u,w — w(Q) form a system of parameters at Q. The
curve E is then defined by u = 0. Similarly, if f, ^0 at g. set w' = t0/t]. Then
u = vw' and u. u;' — w'(Q) form a system of parameters at Q. The curve E is then
defined by v = 0. Hence, J^e/vIeo = ^£o(")_1 and >£/,"|£, = ^£,(^)-1. where
£0 = |2 e £;?o 7^ 0}, E\ = {Q e £;?, ^ 0}. w = w (modJ^J) and U = v
(mod J2"!). So the transition function oIJVeiv on £0 His, is /,o = «/«. In view of
the relation ufo = ut\. we have w/v = to/t\. Thence, we know that JVejv — ^pi (— 1 )•
In particular, (E2) = -1.
(2) With the same notation as above, a* C is defined by a * (/) = 0 near the point
Q of E. Suppose g G iso- Set z = w — w(Q). Then (w.r) is a system of parameters
of V at g. We can express a* f — u^g with g G <^V',g- and g G (^V".g)* if and only
if ffl{\.w{Q)) =£ 0. In fact, the correspondence (u.v) h-> (u. (z + a)u)(a = w(Q))
gives rise to a ring homomorphism a*: &y.p -+ @v.q which is continuous with respect
to the m-adic topology, and we can write g as
g = fM(\.z + a) + uf^+x{\.z + a) + • • • G k[[u.z}}.
In the quotient field of ffv>,g. we have k(V')n&v'.Q = @v.q- (Verify this assertion.)
This implies g G ^V'.g. Clearly, we have the equivalence
ge(0v..e)* iff fM(l.= +a)e(<?v..e)* iff /,(1.a) ^0.
Note that if u\fji{u.v). then fM(l.t) = 0. as an equation of degree ju. has ju roots.

8. INTERSECTION THEORY
171
each root being counted with due multiplicity. By replacing the system of parameters
(u,v) to (u + cv.v) if necessary, we may assume that u\ffl(u.v). By the above
argument, we know that the curve a'C is defined by g = 0 near the point Q. Hence,
a*C = fiE+a'C. Noting here that the hypothesis u\fM(u. v) implies Qx : = (0,1) ¢.
E Ha'C, we have
(C -E) = ]Tdim/t^.e/(«,g)
QeE0
= ]T dim*^.Q/(u.f,,(1. z + a))
QeE0
= /u.
This entails, together with the result in (1) above, that (a*C ■ E) = 0. □
Corollary 8.9. Let C,D be irreducible algebraic curves on V, let ju = /i(C;P),
and let v = ju(D;P). Then we have
(a'C -a'D) = (C ■ D) -/iv and (a'C ■ a* D) = (C • D).
Proof. It suffices to show that (a*C ■ a*D) = (C ■ D). Let f = 0 be a local
defining equation of C at P. With f taken as an element oik(V), set A = C - (f).
Though A is not necessarily an effective divisor. P ¢. Supp(y4) := \Ji A,- i(A = J2i aiAt
is the irreducible decomposition. Noting here that ffv{A) = ffv(C), we have
(cr*C • a*D) = (o*@v(A) ■ o*@v(D)) = (a*A ■ a*D)
= 'Y^ai{a*Ai -a*D).
i
Since o~xAj n E = 0, we have {a*At ■ a*D) = (A, • D). Hence,
[a*C -a*D) = Y^a,i,Ai-D) = (A ■ D) = (C • D). □
i
Let L(C |£>o|) be a linear system on an algebraic surface V with n := dimL >
0. We assume that L has no fixed components and BsL 74 0. Let P £ BsL,
and let a: V —> V be the monoidal transformation with center P. Then a*L : =
{a*D\D e L} is a linear system on V, and the module M(a*L) associated with
a*L is isomorphic to M(L). We call a*L the total transform of L. Let E = o~x{P)
be the exceptional curve. For each D e L, we can write a*D = a'D +/uDE. Let /u =
mmoeLMD- Then/uE is the fixed part of a*L. So we call a'L := {a*D ~/uE;D e L}
the proper transform of L by a. Though a'L has no fixed components, it may still have
base points. Take distinct two members DX,D2 of L without common components.
So (D\ ■ D2) > 0. In particular, if BsL ^ 0, (D\ • D2) > 0. The intersection number
(D\ • D2) depends only on L, not on the choice of two members D\,D2 of L. We
define (L2) = {D{ ■ D2). Then by Corollary 8.9, (a'L)1 = (L2) -M2.
Let f: W —> V be a birational morphism between algebraic surfaces. Suppose
f is a composite of monoidal transformations 07: V,■ —> V\-\ (1 < i < N) such that
Vn = W, Vo= V, and f = o\---on. For a divisor D and a linear system L on F, we
define the proper transforms f'D and f'L inductively as f'D = o'N(o\ ■ ■ ■ aN-\)'D

172
III. ALGEBRAIC SURFACES
and f'L = o'N{o\ ■ ■ • 0-^-1)^- Similarly, we define the total transforms of D and L
by f*D =o*N{o\---oN„x)*D m&f*L = a*N{ox---oN_\)*L.
Lemma 8.10. Let L be a linear system on an algebraic surface V such that dim L >
0 and dim BsL < 0. Then there exists a birational morphism p: W ^> V satisfying
the following conditions:
(1) p is a composite of monoidal transformations with centers at points.
(2)/): W-p~l{BsL) Z F-BsL.
(3) p'L = p*L-(the fixed part of p*L) and Bs(p'L) = 0.
Proof. We continue the arguments before stating Lemma 8.10. If Bs(er'L) ^ 0,
take P\ e Bs(er'L) and let ai: V2 -> V\ := V' be the monoidal transformation
with center P\. If we set o\ = a, we have a^a^L) = {p\o-i)'L, M{a'2a[L) —
M(L), and (<t2<t[L)2 < (^[L)2 < {L2). Meanwhile, since the intersection numbers
(L2), (ff[L)2,... are nonnegative integers, this process terminates after finitely many
times. Then we have only to take as p the composite of the monoidal transformations
considered in these steps. □
Corollary 8.11. Let V be an algebraic surface, and let p: V —> Y be a rational
mapping to a projective algebraic variety Y. Then there exists a birational morphism
t: W —> V satisfying the following conditions:
(1) r is a composite of monoidal transformations with centers at points.
(2) p • r is a morphism.
Proof. We may assume that p is a dominant rational mapping. (Let U be an
open set of V such that U is contained in the domain of definition of p. Replace
Y by p{U).) By (II.5.19). there exists a linear system L on V such that p = ¢/,.
We may assume that L has no fixed components. Then the domain of definition is
V - BsL (II.5.20). Suppose BsL f 0. Take P G BsL. and let a: Vx -> V be the
monoidal transformation with center P. The composite morphism p ■ a: V\ —» Y
is then identified with <JVl. So in view of Lemma 8.10, there exists a birational
morphism r: W —> V such that r is a composite of monoidal transformatioms and
Bs(r'L) = 0. Then p ■ x: W —> Y is a morphism. □
III.8. Problems
1. Let V be an algebraic surface, and let P e V. For an irreducible algebraic curve
C on V with P e C, prove that p.{C\P) = mfPeBi(C.B;P), where B ranges
over all irreducible algebraic curves on V such that B ^ C and P e B.
(Hint. Show first that there are sufficiently many curves B as above which are
nonsingular at P, and utilize it.)
2. Prove that if C is a nonsingular algebraic curve on P2 of degree n, then we have
g(C)=l-(n~\)(n-2).
(Hint. In view of (II.6.22), coc/k — (a>pi/k\c) &>-^c/p2 — ^(n — 3)|c-]
3. Let V be an algebraic surface, let L be a linear system, let /u > 0 be a positive
integer, and let P e V be such that P £ BsL. For a linear subsystem
L-juP:= {D eL;D = S^aiCi,p,{D;P) :=^a^{Cr,P) >ju},
i i

8. INTERSECTION THEORY
173
prove that
dim{L-fxP) > dim/. - -/u(ju + 1).
(Hint. Choose Do e L so that P £ SuppDo. Then the module M associated
with L — /uP is given as
M = {fek(V);D = (/) + Do eL-MP.fe (my.pT}.
If (u, v) is a system of local parameters atP,f£ (m [/p)1" if and only if all coefficients
of the terms of degree 0 up to ju - 1 in the Taylor expansion of f e &v.p vanish.)

CHAPTER 9
Pencils of Curves
Let V be an algebraic surface. A dominant rational mapping p: V —> B from
V to a projective algebraic curve B is called a pencil of curves (a pencil, for short).
We assume that B is nonsingular unless otherwise specified. We call p an irrational
pencil of curves or a rational pencil of curves according as the genus of B is positive
or zero. By (II.5.19), V has a linear system L such that p = <£>l and dimBsL < 0.
Furthermore, by Corollary 8.11. there exists a birational morphism x: W —> V such
that p • r is a morphism. Here p • r: W —> B is a pencil of curves on fF. If there
exists an open set U{^= 0) of B such that Wh is irreducible and reduced for each
b 6 U (we simply say that Wh is irreducible and reduced for "a general point b of
B"), then we call the pencil p: V —> B an irreducible pencil.
Lemma 9.1. Given an algebraic surface V, the following are equivalent:
(1) giving a pencil of curves p: V —> B,
(2) giving a subfieldk{B) of k{V) with tr.deg^k{B) = 1.
Moreover, under the above equivalent set-up, the pencil p is an irreducible pencil if and
only if k(V)/k(B) is a regular extension.
Proof. (1) Given a pencil/): V -> B, p*: k{B) ^k{V) is an injection because/)
is dominant. Hence, k(B) is a subfield of k(V) with tr. deg^ k(B) = 1. Conversely,
given a subfield K of k( V) with tr. deg^ K = I. there exists a nonsingular projective
algebraic curve B such that K = k(B), because K is a finitely generated field over
k. (If we choose an element x of K which is transcendental over k, then we can
consider k(x) = fc(P'). Since K/k(x) is a finite algebraic extension, we have only to
take the normalization B of P1 in K (11.4.23).) The natural inclusion k(B) <-> k( V)
induces a dominant rational mapping p: V —> B such that K = p*(k(B)).
(2) By Corollary 8.11. there exists a birational morphism r: W —> V such that
p ■ t: fF —> 5 is a morphism. Replacing V.p by W7, p • r, we may assume from the
beginning that p is a morphism.
Suppose p is an irreducible pencil. We shall show that k(V)/k(B) is a regular
extension. For this, choose a point Z? of B so that the fiber Vh is irreducible and
reduced, and let & = &Bh. Since ^ is a discrete valuation ring, we denote the maximal
ideal by t&. Choose an arbitrary closed point P of Vh. and let (A. m) = [&v.p- nv./>)-
Then & C A and both A and A/tA are integral domains. We also note that Q{A) =
k(V) and Q{ff) = k(B). We have only to show that Q(A) ®QW K' is a field if K' is
a finite algebraic extension of Q{&). For this, it suffices to show that A ®0 @' is an
integral domain if {&', uff1) is a discrete valuation ring of K' such that &' > &. In
fact, we then have Q{A®g@') = Q{A) ®Q(g) K'. We may take as ff' a localization
of the integral closure of & in K'. We shall show that A' := A ®@ &' is an integral
domain. First of all, note that A is a flat ^-module. (It suffices to show that A has no
175

176
III. ALGEBRAIC SURFACES
torsion as an ^"-module, equivalently in this case that tb = 0 implies b = 0. But this is
clear since A is an integral domain.) Hence, the injection of & -modules u: &' —> &',
zv-*zu induces an injective ^4-homomorphism u: A®g&' ^ A®gt&', a®:^fl®:«.
Namely, u is a nonzero divisor of A' : = A (&g 0'. Suppose b[ ■ c[ = 0 for nonzero
elements b[,c{ of A'. Note that A'/uA' = (A/tA) <g>^ {&' /u@') = A/tA. (In fact,
&'' /u&' - &/t<f - k.) Since A/tA is an integral domain by hypothesis. b[ £ uA' or
c[ g uA'. In case 6[ 6 uA', write 6( = w/^- Then b1-, • c[ = 0 because u is a nonzero
divisor of A'. Repeating this argument, we have b\ e I : = P|«>o u"^' or ci e ^ ^n
the other hand. A' is a local ring with m' := m ® &' + A <g> m^" as the maximal ideal.
(Let ^ be the integral closure of & in K'. Then $" is a localization of &. Since <f
is a finitely generated ^-module (1.1.34). A ®@ & is a finitely generated ^4-module.
Hence, ^4 <g>^ ^ is a semilocal ring, i.e.. a ring with only finitely many maximal ideals,
and the maximal ideals of A ®s & correspond bijectively to the maximal ideals of 0
because A/m <S>A {A ®g ff) = @/t&. A' is a localization of A ®0 & with respect to
one of maximal ideals.) By the lemma of Artin-Rees (1.1.31), we have ul = I. Since
I is a finitely generated ^'-module, if we take a system of generators {x\ x„} of
I. we have x,- ~ ^Z. ua^Xj with a,-7- 6 A'. Let / = det((5,-,- - uctjj). Then fx,- — 0
for each ;' and f - 1 G m^' c m. Hence, / is invertible. and hence, I = (0). So ^4'
is an integral domain.
(3) Conversely, we shall show that p: V —► B is an irreducible pencil if Ar( V)/k(B)
is a regular extension. As in (2), we may assume that p is a morphism. Furthermore,
in order to simplify the subsequent argument we assume that the characteristic of
k is zero. Choose an affine open set U of B and affine open sets {P/}'=1 of V
so that p~x{U) = {Jj=l Vj. Consider pt : = p\Vi: V, —> B. If we can choose an
open set U,- of B so that (Vj )b is irreducible and reduced for each b e U/. Vb is
irreducible and reduced for each b e U n (fXi=l Uj). Hence, replacing V. B by V,-, U,
respectively, we may assume that V = Spec(i?) and U = Spec(S). Set K = Q(S).
Since Rk '■= R®s K is a finitely generated domain over K, Noether's normalization
theorem (1.1.15) implies that there exists an element t\ e R such that RK is an
integral extension of K[t\\. Since the characteristic of k is zero, Q(R) is a separable
algebraic extension of K{ti). So we may write Q(R) = K{tx. h). \,ztF{t\,X) be the
minimal polynomial of t2 over K{t\). Replacing t2 by an element of the form g(t\)t2
with g(t\) e S[t\], we may assume that t2 is integral over S\t\\ Then t2 € Q(R)
is integral over R, and since R is a normal ring. t2 € R. Replacing U by its open
set if necessary, we may assume that R D S[t\.t2], R is an integral extension of
S[ti.t2], Q{R) = Q(S[tut2]). and S[tut2] = S[7V r2]/(F(rb T2)). We shall prove
the following result. □
Assertion 9.1.1. There exists an open set U of B = Spec(S') such that
Fk(b)(Ti. T2) is an irreducible polynomial over k(= k{b)), where Fk^(T\, T2) is
obtained from F(T\, T2) by replacing the coefficients by their images in k(b) = @B.h/w-B.b-
Proof. Write F(TUT2) = Y.ac<*Ta, a = (ai.a2), Ta = T"'T22, ca e S,
\a\ < d (d = degF). Replacing S by a quotient ring of the form S[\/a], we may
assume that ca ^ 0 implies (cQ)_1 £ S. Hence, degiJ't(fc)(7,1. T2) = d for each
b G B. Fix a pair of positive integers (p,q) with d = p + q. Consider a set of
indeterminates {7^, J";/? = {P\,Pi).y = (71,72), |/?| </>, \y\ < q} and a polynomial
ring C = 5[Jg, T"; \fi\ < p.\y\ < q] over S generated by these indeterminates.
Let a be the ideal of C generated by a collection of elements of C, {Pa(TL T") : =

9. PENCILS OF CURVES
177
Y,p+y=a TpT" - cQ; |a| < d}, and let Q. be an algebraic closure of K = Q(S). Then
we have the equivalence of the following two conditions:
For some F\,F2 € Q[T\, T2], degFj = p, degF2 = q. F = F\ • F2 if and only if the
closed set V(a <% K) of Spec(C <g>s K) is not empty.
Similarly, we have the following equivalence
<?!. G2 G k(b)[Ti.T2]. degG! = p. degG2 = q.Fk{h) = G{ ■ G2 if
and only if the closed set V(a <S>s k{b)) of Spec(C <% k{b)) is not
empty.
Since F{T\. T2) is an irreducible polynomial in £l[T\. T2]. we have V(a<S>s K) =
0. (In fact, since k(V)/k(B) is a regular extension. k(V) ®k(B) ^ is an integral
domain and Q{k(V) ®k{B) Q) = Q{Q.[TU T2]/{F(Ti.T2))). Hence. F{TX.T2) is an
irreducible element of 0,[TU T2]. Hence, a <% K = C <g>s K. So 1 = 5Zui<rf haPa
with ha £ C <S>s K. Let a be a nonzero element of S such that aha € C for each
a, and let £/^ - Spec>S[a_1]. Then V(a<8>s k{b)) - 0 whenever b e U^q). If we
set t/ = Op+q=d U(p.<i)< tne assertion holds true with this U. D
Assertion 9.1.2. There exists an open set U of B such that Vb is irreducible for
each b £ U.
Proof. Note that R is a normal ring, R D S[t\. t2] is an integral extension, and
Q(R) = Q{S[t\,t2]). Choose an open set D{h) of SpecSfr, t2] (h G S[ti,t2]) so
that D{h) nSing(Spec5[fi.f2]) = 0- Then S[ti.t2.h~l] = R[h'xl (S[tu t2. h~l] is
a normal ring, and R[h~l] D S[t\.t2-h~l] is an integral extension.) By virtue of
Assertion 9.1.1, we can take an open set U of B satisfying the condition:
k(b)[tu t2] = k{b)[T\.T2]/{Fk(h){Tu T2)) is an integral domain for
each b € U.
Here we may assume that the following condition holds:
hk(h) (= the image of h in k{b)[tu t2]) ^ 0 for each b £ U.
Then it follows that k{b)[tx,t2, {hk(h))~l] = Rk(k)[(hk{b))~l] for each ft e V', where
Rk{h) = R®s k(b). For the closed set V(h) of Spec(i?), we have dim VK(hK) < 0,
where VK(hK) = V(h) <g>s K. Hence, replacing U by a smaller open set, we may
assume that dimV{h) <g>s k(b) < 0 for each b e U. (In fact, for an S-algebra
R = R/{h), R ®s K is an artinian ring by hypothesis. Hence, R ®s K is a finitely
generated AT-module. Then we can find a € S so that R <g>s -^[a-1] is a finitely
generated S[a^-module. We have only to take SpecSja-1] instead of U.) Namely,
for each b e U, Sjxck(b)[t\. t2. (hk^)~l] is a dense open set of Spec/?t(/>) an<i it
is also irreducible and reduced. This implies that Vb = SpecRk(h) is irreducible for
each b e U. (In fact, Rk{b) = R <g>s &/t{R <8>5 &), where & = 0B,h and W = mB,h.
Here R <g)S & is a normal ring as a quotient ring of i?. By (1.1.33), each prime divisor
of t (R <s>s @) has height 1 and corresponds to an irreducible component of Vf,. So
each irreducible component of Vb has dimension 1.) □
Assertion 9.1.3. There exists an open set U ofB such that Vh is reduced for each
b€U.

178
III. ALGEBRAIC SURFACES
Proof. Look at a sequence of ring extensions S[t[] c S[t\. ti\ C R, where we
may assume that R is an integral extension of S[t\]. Hence, R is a finitely generated
>S[?i]-module. and RK is a finitely generated ^T[?i]-module. Since RK is a torsion-
free ^T[?i]-module and K{t\] is a principal ideal domain. RK is a free ^T[?i]-module.
Replacing 5 by a quotient ring of the form S[a~'] if necessary, we may assume that
R is a free >S[?i]-module. This entails that Rk(h) is a free fc(6)[fi]-module for each
b e B. Then Rk{h) is a subring of R'k{h) : = Rk(h) ®k{h)[t]] k{b){t\). Hence, if R'm is
reduced, so is Rk(h). By Assertion 9.1.2. we may assume that Vh is irreducible for
each b e B. Then R'k,h) is a fc(&)(*i)-algebra which is an artinian local ring. Since
the characteristic of k is zero, we have the equivalence:
R'k{h) is reduced if and only if Q.R,^/k{h){ll) = D.R/S[ll]®k(b){t\) = (0)
(compare Problem III.9.1). Here QR/s[,t] is a finitely generated S^j-module, and
nR/s[t]]®K(h) = (0). (In fact. R®S[tl]K(ti) = Q(R)- and Q(R)/K(tx)I is a separable
algebraic extension.) Hence, there exists a nonzero element g(?i) £ .S[?i] such that
g(^i)^/^,,] = (0). If we choose an open set U of B so that gk(h)(ti) ^ 0 for each
b g £/, it follows that F/, is reduced for each b e U.
This completes the proof of Lemma 9.1. □
Let p: V —> 5 be a pencil of curves such that p is a morphism. With K : = k(B)
identified with a subfield ofk{V). let ^ be the algebraic closure of K in k(V). K is
a finitely generated extension of k with tr. degj. K = 1 (1.1.5). Hence, there exist a
nonsingular projective algebraic curve B and a finite morphism v: B —> B such that
p: V —> 5 is a composite of rational mappings
/>: V '^B^B
and p*k(B) coincides with the subfield ^ of k(V). We call this decomposition of
p the Stein factorization of p. Here /5 is a morphism. (In fact, take affine open sets
Speci? and SpecS of V and B. respectively, so that p~l{SpecS) ^ SpecR. Let S
be the integral closure of S in K. Then SpecS = v~'(SpecS). Furthermore, S C R
because R is a normal ring. Thence. /?|sPec/^ Speci? —> SpecS is a morphism.)
Corollary 9.2. Suppose the characteristic of k is zero, p is then an irreducible
pencil of curves. lfB ^ B. a sufficiently general fiber of p has deg v (:= [k(B): k(B)])
connected components. (Even if the characteristic of k is not zero, each fiber of p is
connected.)
Proof. Ifthe characteristic of k is zero, k(V)/k(B) is a regular extension (1.1.8).
We can than apply Lemma 9.1 to prove the assertion. We shall not give a proof for
the parenthesized assertion. □
Lemma 9.3 (Stein factorization). Let p: V —> B be a pencil of curves such that
p is a morphism. Then the following assertions hold:
(1) If p*ffv = &b, then every fiber of p is connected.
(2) If k(B) is algebraically closed in k{V), then we have p^&v = ffB.
Proof. (1) Let b be a closed point of B. & = efBj, is a discrete valuation ring.
We let t be a generator of m^. Replacing V. B by V xB Spec^, Spec^, respectively,
we have only to show that the fiber of p over the closed point of Spec^ is connected

9. PENCILS OF CURVES
179
when B = Specif. For an integer r > 0, let Vr = V ®ff ff/(tr+x), where F0 is the
closed fiber. Suppose Fo is a sum of n connected components. Then T{VQ,ffVa) has
idempotent decomposition of the unity
1=^)+^(0) + ... + ^0).
Namely, ef] e T{V0,ffV(1), {ef])2 = e,(0)(l < i < n) and ef] ■ ef = 0 whenever
i ^ j. In simple terms, if Fo = Fn ]J ■ • • ]J F)„ is the decomposition into connected
components, ef' is regular function on Fo which is 1 if restricted on Fo, and 0 if
restricted on VojU ^ i)- Since Vr is homeomorphic to F0, Vr has n connected
components as well. Hence, the unity element 1 oiY(Vr,@vr) is decomposed into a
sum of idempotents
1 = e\r) + eW + • • • + e<r)
such that e\s) = e\r) (mod?s+^)(1 < i < n) if s < r. We shall show that the
assumption n > 1 leads to a contradiction. For 0 < s < r, we have the following
commutative diagram:
0 ^t&V *&y-^0Vo ^0
no, &.
0 ^ ts+x@v 5» @v —^ &v< >- 0
A ,,
V*r (,r
0 >tr+x@v ^@v—PjL^@vr ^0
where rjsr is the natural injection tr+xffv «-> ts+xffv and the other homomorphisms
are the natural residue homomorphisms. From this diagram, we obtain the following
commutative diagram of cohomology sequence,
0 *~T{V,t0v) *T{y,<9v)-^T(VQ,<9Vl>) *~Hx(y,t6v)
A
fOs l'o,
(*) 0 ^T{V,ts+x@v) *~T(V,0v)—^T{ys,0vt) *-Hx{V,t'+l0v)
i' A
o—>-T{v,tr+x@v)—^r(F<^v)-^r(Fr,^vr)—*-Hx{v,tr+x@v)
where ns,(isr, vsr are the homomorphisms induced naturally by ps, rjsr, £sr. Here note
that T{V,ts+x0v) = ts+xY{V,&v) and Hx{V,ts+xffv) = ts+xHx{V,ffv) for each
s > 0. Furthermore, the hypothesis implies that T{V,@V) = if, and Ex{V,0v) is a
finitely generated if-module (II.4.18). Since if is a principal ideal domain, we can
write
HX{V,@V)=F®T, F^ff®m

180
III. ALGEBRAIC SURFACES
The commutative diagram (*) can be rewritten as follows:
0 ^ff/tff ^r(F0,dV0) *tHx{V.<9v)
A A
0 ^ff/ts+xff-^*Y{Vs, ffVt) *ts+xHx{V,0v)
T T T
r
0—^^//r+1^-^-^r(Fr,^r) *tr+xHx(V,0v).
Here CokerTt, C tr+xT since f+1r(Fr,dVr) = (0). Hence, Coker^ = (0) if r > au
So 1111(¾) = Im(vs,) for r > 0. Here since Imfo,.) = @/ts+x@^ we have ef' Im(^) 7^
(0) for some i and ey Im(7t4.) = (0) for all j ^ / for the idempotent decomposition
of the unity 1 = e[s) + e^ + ■■■ + 4'v) in T{Vs.0vJ. Meanwhile, ej5)lim(v„) = ej°
(modts+l&yr) ^ (0) for each j. This is a contradiction if n > 1. So Vq is connected.
(2) We shall show that p*&v = ffB if k{B) is algebraically closed in k{V). Since
P+&V is a coherent &s-Module. T(p~x (U), dV) is a finitely generated S-module if t/ =
Spec(S) is an affine open set of B. We have T{p-x{U),ffv) c r(/7->(^),MF)) =
fc(F). Each element dj of T(/>~'(£/), $V) is integral over S, and £, as an element of
/c(K), is therefore algebraic over 0(S) = k(B). Hence, £, e Q(5). Since d; is integral
over S and 5* is a normal ring, we have £, € S. Hence, T{p~x{U),&v) ~ S. So
p*&v =@s- □
Remark 9.4. Suppose, for a pencil of curves p: F —» 5, /? is a morphism and
k{B) is algebraically closed in k{V). Let 6 be a sufficiently general, closed point of
B, and let t be a local parameter at ft. t being regarded as an element of k(V), we
view the fiber V/, = V <g># fc(ft) the divisor (?)o (= the zero part of (t)) on F. If the
characteristic of k is zero, then (f)o is an irreducible divisor by Lemma 9.1. If the
characteristicp of k is positive, then we can write (?)o = p"C [a > 0, C: irreducible
component), though we omit a proof.
We suppose a linear system L is given on an algebraic surface F and n : =
dim L > 0, dim Bs L < 0. Let <D be the rational mapping <DL: F —> P" associated
with L. For the domain of definition U of ¢, we denote by 4>(F) or Im<D the
closure of <S>(U) in P" and call it the image of V by ¢. Since n = dimL, there is no
hyperplane containing O(F). Hence, putting f = <S>\u, the natural homomorphism
/i: H°(Pn,@P„(l)) -► H°{U,tp*@p,{l)) is injective. Let & = &V(D) with D e L.
Then <$*&?* (1) = Se\ u. By hypothesis, F - t/ is at most a finite set, and accordingly,
H°(U,3'\U) = H0(V,2') (compare the proof of (11.5.19(2))). Hence, ft gives rise to
a homomorphism //°(P".^P»(1)) -> H*(V,£?). For a e jff°(P",^(l))(ff f 0), we
denote by H(a) the hyperplane of P" defined by a and by Z>(<x) (|u(ct))o or F • H{a))
the effective divisor on F defined by p{a).
If h > 0, then we have dim<D(F) > 1. If dimO(F) = 1, then <D factors as
4): T/..f^5^p". B:=®(V).
Though 5 is not necessarily nonsingular, the rational mapping p: V ■ ■ • —* B is a
pencil of curves. We then say that the linear system L is composed of the pencil p.

9. PENCILS OF CURVES
181
k{B) is identified with a subfield of k( V). Take a nonsingular projective algebraic
curve B so that k(B) is the algebraic closure of k(B) in k(V) and factor p as
p: V ■■■^B^B.
where v: B —> 5 is the normalization morphism. The homomorphism p. therefore
decomposes as follows:
p: H°{P\^{1)) ^ HQ{B,ffB{\)) ^ H°{B,ffs(l)) C H°(V,&),
where &B{\) = (^p»(l) ®&b and ^(1) = v*<fB(l). Let J = deg^(l) and AT =
degv := [k(B): k(B)]. Then for a € i/°(P".^P»(1)), we can write (v*z*(er))0 =
Y,?=\ bi. For ft e B, we define a divisor F^ of V as follows:
Let U be the domain of definition of p as before, and let t be a local parameter
of B at ft. We define F^, as the closure £\ a7C, of the divisor (?)o = £) ■ ajcj on
t/.
If/5 is a morphism, V-h is identified with the fiber of p at the point ft. For the
above a, D{a) = Y^=\ V~b- ^or a Pencn °f curves p: V —> 5 in general and a point
ft G i?, we can define the divisor F/, of F in the same fashion as above.
Lemma 9.5. For a pencil of curves p: V —► B, the following assertions hold:
(1) If p is an irrational pencil, then p is a morphism.
(2) If p is a rational pencil, then { F/,; ft G B} is a linear system of dimension one.
Proof. If p is not a morphism, there exists, by Corollary 8.11, a birational
morphism t : W —> F such that /> • r is a morphism.
(1) Suppose /> is not a morphism. The morphism r is given as a composite of
monoidal transformations with centers at points. So decomposing r as
we may assume that p • xi is not a morphism and x\ is a monoidal transformation
with center at P e F'. Let £ = z^\P). By (II.5.19), there exists a linear system L
on V' such that dimBsL < 0 and p • r2 = ¢/.. Then /1 is a base point of L, and
r • p = <!>£<, where L' is the proper transform of L by ^. Let// = mmDeL p(D;P).
If £> is a sufficiently general member of L, r[D : = t*(D) — pE does not contain E
as an irreducible component and (r[D • E) = p. > 0. Let D' G Z/. Then the image
of M(L') by the natural homomorphism H°(W,ffw{D')) -> H°{E.@E(D')) defines
a linear system Tr£(L') on £, where ^(D') = &W{D') ®ffw &e- Since BsZ/ = 0,
Bs(Tr£(L')) = 0 and the rational mapping defined by Tre(L') coincides with Q>l'\e
(compare the proof of Lemma 8.6). (p • r)\E: E —> B is therefore a dominant
morphism. Since £ is a nonsingular rational curve, B is also a rational curve by
Luroth's theorem (1.1.36). This contradicts the hypothesis. So p is a morphism.
(2) Let fti,ft2 be any two points of B. Identifying B with P1, we can choose
an inhomogeneous coordinate t of P1 in such a way that b\ and b2 are respectively
defined by ? = 0andf_1 =0. Thenft,-ft2 = (?)• Clearly, (t)0= Vh] and (f-1 ) = Vhl.
Hence, F/,, ~ F/,2 (linear equivalence). D
Conversely, let L be a linear system on F such that dim L = 1 and dim Bs L < 0.
The rational mapping ¢^: F —> P1 is then dominant. Namely, 4>£ is a rational pencil.
Below, we call a one-dimensional linear system, which may have fixed components, a

182
III. ALGEBRAIC SURFACES
linear pencil. The next purpose is to show that first theorem of Bertini {Irreducibility
theorem).
Theorem 9.6 (The first theorem of Bertini). Suppose the characteristic of k is
zero. Let V be an algebraic surface, and let L be a linear system on V such that
dimL > 0 and L has no fixed components. IfL is not composed of a pencil of curves,
then a general member of L is irreducible and reduced.
We reduce the proof of the theorem to the subsequent Lemmas 9.7 and 9.8. The
hypothesis that the linear system L is not composed of a pencil of curves implies
that N : = dim!, > 2 and dimOL(F) = 2. Let {f0, ■ ■ ■ .fN} be a fc-basis of M(L).
Then Ol: V -► P^ is given by P ^ (/o(-P)...., fN{P)). Hence, k{<t>L{V)) is the
subfield of k( V) generated by f\ //o, — /;v//o over k. Here we may assume that
/i //<)• /2//0 are transcendental over k. For c ek*. let Mc = k{f\ + c/2) + fc/o be
a subspace of M(L), and let Ac be the linear subsystem of L whose A:-module is Mc.
Ac is a linear pencil, and £($a, (^)) coincides with k((f\ + ¢/2)//0). Replacing
/1 by an element of the form f\ + 02/2 H h a^fN (<*/ e fc) if necessary, we may
assume that Ar has no fixed components if c is a general element of k*. We shall
assume this condition. If we can show that k{{f\ + ¢/2)//0) is algebraically closed
in fc(F) for a general element c of k*, a general member of Af is an irreducible and
reduced divisor by Lemma 9.1. Hence, a general member of L is irreducible and
reduced as well. To complete a proof of Theorem 9.6, we have only to prove the
following result.
Lemma 9.7. Let k be afield of characteristic zero, and let K be a finitely generated
field extension ofk such that r := tr. deg^ K > 0 and k is algebraically closed in K.
Let x\,..., xp (1 < p < r) be a set of elements of K which are algebraically independent
over k, and let x,- = c(\X\ +•••-)- cipxp (1 < i < p;c;j G k). Then there exists a
(p — 1) x p-matrix (c,;) with entries in k such that tr.deg^£:(^ ,xp-\) = p — \
and k{x\,... ,xp-\) is algebraically closed in K.
For a proof of Theorem 9.6, we put K = k{V), p = 2, and x,- = /,//0 (i =1,2)
and apply Lemma 9.7. For a proof of Lemma 9.7, we need the following lemma.
Lemma 9.8. Let k be a field, let K be a field extension of k, and let x be an
element which is transcendental over K. Ifk is algebraically closed in K, then k(x) is
algebraically closed in K{x).
Proof. Let f(x),g(x) be mutually prime elements of K[x\. We shall show that
if t := f(x)/g(x) is algebraic over k(x), then t e k(x). If t e K, then t is algebraic
over k. (In fact, otherwise, the minimal polynomial of t over k(x) has a coefficient
not in k. Then x is algebraic over K which contradicts the hypothesis.) Hence,
t e k. Suppose t $lK. Then it is clear that t is transcendental over k. Considering
an algebraic relation of t over k(x), we see that x is algebraic over k(t). Now write
f(x) = a0x" + V a„, g(x) = box" + \-b„,
n > 0, at,bj e K. a0 ^ 0 or b0 ^ 0.
Then x is a root of an algebraic equation with coefficients in K{t),
(*) g(X)t - f{X) = (b0t - a0)X" + ■■■ + {bnt - an) = 0.
This equation is an irreducible equation over K{t). (If it is reducible, there exist

9. PENCILS OF CURVES
183
elements F(t. X). G{t. X) of K[t. X] and a nonzero element (p{t) of K[t] such that
0 <degxF <n, F(t.x) = 0, and
<p(t) ■ ((b0t - a0)Xn +■■■ + (bj - a„)) = F{t.X)G{t. X).
Meanwhile, since t - f(X)/g(X) is a prime element of K(X)[t] and f(X), g(X) are
mutually prime, g(X)t - f{X) is a prime element of K[t. X]. (This is an argument
used to show that if t is an indeterminate, R: UFD implies R[t]: UFD.) Hence,
either degx F = n or deg^ G = n. This contradicts the choice of F.) The set of n
roots of the equation (*) is a part of the set of conjugate roots of x over k(t). Hence,
the coefficients £,• := (btt - a,)/(b0t - a0)(\ < i < n) of the minimal equation of x
over K(t) are algebraic over k{t). Replacing t by t~l if necessary, we may assume
Z>o ^ 0. Suppose £,• satisfies an irreducible equation over k(t).
(t) ^+71(0^ + --- + ^(0 = 0. for each Yj(t) e k(t).
where N depends on i. If we take elements c0(t),... ,cN{t) of k[t] such that c0(t),...,
cN(t) have no common factors and yj(t) = cj{t)/co(t) the equation (f) is rewritten
as follows:
(t|) c0{t)(bjt - aif + cx{t){bit - fl,-)""1^ - ao) + ■ ■ • + cN(t)(b0t - a0)N = 0.
Since t is transcendental over K, (t|) is a polynomial identity. Summing up in (t|)
the coefficients of terms of highest degree in t and dividing the sum by a power of bo,
we obtain a nontrivial algebraic equation of bi/bo over k. Since bj/bo e K and k is
algebraically closed in K, we have bj/bo 6 k. Thus, g{x) = bog] (x) with g\ (x) G k[x\.
If aobo y^ 0, we can repeat the above argument with t replaced by t~l to conclude
that f{x) = aofi(x) with fx{x) G k[x]. Then t = {ao/bQ){f\{x)/g\{x)) and
f\{x)/g\{x) e k(x). Hence, a0/bo is algebraic over k(x). The observation, made in
the case t £ K, implies ao/bo £ k. Consequently, t e k(x).
If flo = 0. then replace t by t + c (c G k*). Then t + c = (eg + f)/g, and
degg(x) = deg(cg(x)+/(x)) = n. In view of the case aobo ^ 0, we have t + c ek(x).
Hence, t <E k{x). □
Proof of Lemma 9.7. We treat only the case p = 2. Let K' be the algebraic
closure of k{x\.xi) in K. If k{c\X\ + c\x2) is algebraically closed in K'. then it is in
K as well. Thus, we may assume that K = K' and r = 2. Then K is an algebraic
extension ofk{x\,x2). In the subsequent arguments, we fix an algebraic closure K
of K. Given a subfield L of K, we take an algebraic closure L of L as a subfield
of K. So L is algebraically closed in K if and only if K n L — L. For c e k, set
x = xi + cx2 and Ot = k(x) n /^. We shall show that Of = k(x) unless c belongs
to a finite subset of k. Set Kc = Of(x2). Then k{x\,x2) C /:,. C /:, and Kc is
an intermediate extension of a separable algebraic extension Kjk{x\.xi). There are
only finitely many intermediate extensions of this kind. (In fact, by the abuse of
notation, let K/k be a separable algebraic extension, let L/k be a minimal normal
extension containing K, and let G = Gal{L/k). Then an intermediate field M of
K/k is given as the invariant subfield of L with respect to a subgroup G(M) of G
(Theorem of Galois). Meanwhile, there are only finitely many subgroups of a finite
group. Hence, there are only finitely many intermediate fields of K/k.) Hence, there
exists a finite subset S (might be an empty set) of k such that, for each c e k — S,
there is a. d € k - S, c ^ d, such that Kc = Kd. Given such c,d,we shall show that
£V = k(x). Set K* = Kc = Kd. K* = nd(x2) = ild(xi + cx2), and K* is a purely

184
III. ALGEBRAIC SURFACES
transcendental extension of ilj. The hypothesis k OK — k implies knild = k. So
by Lemma 9.8, k(x\ + 0x2) = k(x\ + cx2) PiK*. Meanwhile, since D.c is algebraic
over k(x\ + cxj), we see that Q, = k(x\ + cx2) — k{x). D
Let /?: F -> 5 be a pencil of curves on an algebraic surface V. We say that p is
afibration if p is a morphism and a general fiber of p is irreducible and reduced. We
call the fiber over a closed point of B a dosed fiber of p (or simply, a fiber), and the
fiber over the generic point of B the generic fiber of p. There might appear fibers of
p which are reducible or nonreduced.
We shall next prove that in case the characteristic of k is zero, a general fiber
of a fibration p: V —► B on an algebraic surface V is a nonsingular algebraic curve.
For a field K. we denote by K an algebraic closure of K. A local ring (/?.m) is
called a geometric local ring over K if it is a localization of a finitely generated K-
algebra. Given a geometric local ring R over K. we call R an absolutely normal ring
(or a universally normal ring) if both R and R®K K are normal rings. A ring with
only finitely many maximal ideals is called a semilocal ring. The intersection of all
maximal ideals of a ring is the Jacobson radical. From Lemma 9.9 to Theorem 9.11
below, we restrict ourselves to the case where the ground field has characteristic zero.
Lemma 9.9. Let K be afield of characteristic zero, and let (R, m) be a geometric
local domain of dimension one defined over K. Suppose the residue field M of R is an
algebraic extension of K. Then the following assertions hold:
(1) R ®jc K has exactly [M: K] maximal ideals.
(2) If Q(R)/K is a regular extension, then the next three conditions are equivalent
to each other:
(i) R is a normal ring.
(ii) R is an absolutely normal ring.
(iii) Qfl/jf is a free R-module of rank 1.
Proof. (1) By hypothesis, we can express M = K[x]/(f(x)). Let K' be the
decomposition field of the irreducible polynomial f(x) which we take as a subfield
of K. Then M ®K K' = M x ■ • ■ x Nr(r = [M: K]) and M <= K' for each 1. Since
by (1.1.18), there is a one-to-one correspondence between maximal ideals of R ®k K'
and maximal ideals of M ®K K', R ®K K' is a semilocal ring with exactly r maximal
ideals {m],..., mr}, and R ®k K'/vcij = N, and f),=1 m,- = m <s>k K'.
(2) (i) implies (ii). Let K' = K in (1). Note that R ®K K' is an integral domain
by hypothesis. In order to show that R (¾ K' is a normal ring, it suffices to show
that (R <&k K')m, is a normal ring for every m,. Meanwhile, writing m = tR, we
have mi{R ®K K')m, = m(R ®K K')m, = t{R ®K K')mr Thus, (R ®K K')m, is a
normal ring.
(ii) implies (i). Since R is a geometric local ring, we can write R = Ap for
a finitely generated /^-algebra domain A and a maximal ideal p of A. Let A be
the integral closure of A in Q(A). Then R := A ®A R is the integral closure of
R in Q(R). By (1.1.34), A is a finitely generated A-modu\e. Hence, R is a finitely
generated .R-module. So R/R is a finitely generated .R-module. Since R ®k K is a
finitely generated R ®k ^-module, the hypothesis yields R ®K K = R ®K K. Hence,
(R/R) ®K K = 0. This entails R = R.

9. PENCILS OF CURVES
185
(i) implies (iii). m/m2 is an M-vector space of rank 1. This implies that in an
exact sequence of K -modules
m/m2 -> O.R/K ®RM -> O-m/k -> 0.
fi? is an injection. Moreover, since M/K is a separable algebraic extension. D.M/K =
(0). Hence, CIr/k ®rM = m/m2. Applying Nakayama's lemma, we know that Qr/k
is a free J?-module of rank 1.
(iii) implies (ii). Hr^^/k is a free -^ ®* ^-module of rank 1. Hence, for every
maximal ideal n of R ®K K, H^RtSKx)„/K is a ^ree (-^ ®* ^On-module of rank 1. So
it suffices to show that, under the assumption M = K. condition (iii) implies that
J? is a normal ring. Then by (II.6.20). J? is a regular local ring. Since dim/? = 1.
R is a discrete valuation ring. R is therefore a normal ring. □
Lemma 9.10. Suppose the characteristic ofk is zero. Let p: V —> B be a fibration
on an algebraic surface V. Let n be the generic point of B, let K = k{B), and let
Vn = V xB Spec(^T). (We call Vn the generic fiber of p.) Then the following assertions
hold:
(1) Vn is a normal complete algebraic curve defined over K. Moreover, Vn ®k K
is a nonsingular complete algebraic curve defined over K.
(2) There exists an open set U(^ 0) of B such that Vy, is a nonsingular complete
algebraic curve defined over k for each b e U.
Proof. (1) It is clear that Vn is a complete algebraic curve defined over K. The
function field K{Vn) is equal to k(V). and K{Vn)/K is a regular extension. (In fact,
we can make use of Lemma 9.1 because p is a fibration.) Let P be a closed point
of Vn, and let Q be a closed point of V such that Q e {P}. Then the local ring
&v,.p = &v.p is a quotient ring of the local ring @v.q< Here &v.q is a normal ring
as it is a regular local ring. Hence. &vn.p is a normal ring. Vn is. therefore, normal.
By means of Lemma 9.9, Vn ®k K is normal as well. Then Vn ®k K is nonsingular
as an algebraic curve defined over K.
(2) By Lemma 9.9 and (1) above, ilyjK is an invertible sheaf on Vn. We shall
show that there exists an open set U(^= 0) of B such that £V/fll/>-'(£/) is an invertible
sheaf on p~x{U). Let U — Spec(^4) be an arbitrary affine open set of B, and
let {V,- = Spec(Rj); 1 < i < n) be a finite affine open covering of p~l(U), i.e.,
p~l(U) = \J"=l Vj. Replacing U and V/ by smaller open sets if necessary, we may
assume that ClRi/A <%u K is a free (R, ®A K)-module of rank 1. Take e e Qr,/a so that
e ® 1 is a free basis of D.r^a ®a K. Then £lRi/A/Rte is a finitely generated 7?,-module
and (SIri/a/Ric) <giA K = (0). Hence, there exists a nonzero element a, of A such
that O.R./A ®A A.a7X\ — Rt ®a A{a~x\ Set a = a, • • ■ a„, and set U = Spec4[a-1]
anew. ThenQK/B|/;-i(c/) is aninvertible sheaf onp~](U). Set W = p~l(U). We may
assume that Vh is irreducible and reduced for each be U. Q.W/u\vb is an invertible
sheaf on Vt- For each P e Vh, set R = &vb.p- Then D.Vb/kP = O.R/k is a free R-
module of rank 1. R is, therefore, a regular local ring by Lemma 9.9. Namely, Vh
is a nonsingular algebraic curve. □
The following result is called the second theorem of Bertini.
Theorem 9.11. Suppose the ground field k has characteristic zero. Let V be an
algebraic surface, and let Lbea linear system on V such that dim L > 0 and dim Bs L <
0. Then a general member of L is nonsingular outside the set BsL.

186
III. ALGEBRAIC SURFACES
Proof. By Lemma 8.10, there exists a birational morphism r: W —> V such
that Bs(r'L) = 0 and t: W - t_1(BsL) ^ V - BsL. Hence, if D is a general
member of L, t'D - r~'(BsL) is isomorphic to D - BsL. So replacing V by W,
we may assume that BsL = 0. Consider, first of all, the case d\m<bL(V) = 1. Then
there exists a fibration <p: V —> B such that a general member D of L is written
as D = J2Cj=\ Nh, (h,- 6 B), where we may assume that Z>,'s are sufficiently general
points of B and no points of the 6,'s appear repeatedly. (In fact, with the notation
preceding Lemma 9.5, ¢/,: V —> P" has the Stein factorization
and D is written as the pullback V • H of a hyperplane /L of P". If H is sufficiently
general, B HH consists of distinct d points z\ =d of B and v_l (z,) consists of
// distinct points for each -, (compare (11.6.28).) By Lemma 9.10, each Vh. is a
nonsingular algebraic curve. (As in this case, even for a reducible algebraic scheme
X, we say that X is nonsingular at a point x if the local ring &x.x is a regular local
ring.)
Consider next the case dimO^F) = 2. Let {/0.... ,/„} (/,• G k{V)) be a k-
basis of M(L). As L C |D0|. we assume that /o = 1 by taking D0 so that D0 e L. For
A = (X0, ■ ■ ■ ,Xn) G k" (= {closed points of A"}), let A^ be the linear subsystem of L
corresponding to a linear subspace k • 1 + k • (^if\ H (- /l„/„) of M{L). Setting
DA = Do + (1 + /li /i H + /l„/„), we call A;, the linear pencil generated by Do and
Dx. By hypothesis, tr. deg^. k(f\ , /„) = 2. Hence, if X is general point of k",
k(A\f\ H VKfn) is algebraically closed in k(V) (compare the proof of Theorem
9.6). If we replace X by cX = (cX\,..., cXn) (c G k*). Dx is an irreducible and reduced
algebraic curve and DA is nonsingular outside the points of Bs(A/i) = SuppD0 n
SuppD; (Problem III.9.3) by the first case considered above. Since BsL = 0. there
exists a member D\ of L such that SuppD0 D SuppD] n SuppD;. = 0. Writing
D\ = Do + (g), we may assume that k{g,X\f\ + • • • + Xnf„) has transcendence
degree 2 over k. Then we can find a e k* so that k((l + aX\f\ H 1- aX„f„)/g) is
algebraically closed in k{v) (Lemma 9.7). We may view DA as a general member of
the linear pencil generated by D\ and Do + (1 + aXxfx -\ h aXnf„). Hence, DA is
nonsingular outside SuppD0 n SuppD]. D; is, therefore, a nonsingular irreducible
algebraic curve. □
Let p: V —f B be a fibration on an algebraic surface. For P e V and b = p(P),
&B.b is a principal ideal domain and &v.p is a torsion-free &g.b -module. So &VP is
a flat &s.b-module. Under the situation of Lemma 9.10 (2), p: p~l(U) —> U is a
smooth morphism. When the characteristic of k is positive, there exists a fibration
for which such an open set does not exist. However, by the proof of Lemma 9.1, we
know that there exists an open set U of B such that each fiber of p: p~l(U) —> U is
irreducible and reduced. Since B - U is a finite set, there are, in the fibration, at most
finitely many reducible fibers or nonreduced fibers. For a point b of B, if we choose a
local parameter t of B at b and regard t as a function on V, Vy, is viewed as the zero
part (?)o in an open neighbourhood of Vh. Namely, we have Vh = (?)o — S/=i aiQ
(a, > 0, C,: irreducible component) as divisors in an open neighbourhood of V/,.
We call Vh a singular fiber unless it is a nonsingular irreducible algebraic curve. We
set a — gcd(a\,. ..,ar) and call it the multiplcity of Vh. If a > 1, we say that Vh is
a multiple fiber.

9. PENCILS OF CURVES
187
Definition 9.12. Let V be an algebraic surface, let C be a nonsingular projective
algebraic curve, let 21 be an invertible sheaf on V x C. let P, g be closed points of
C, and let q: VxC -► C be the projection. Then q~l(P) = V x (P) and q~l{Q) =
V x (g) are identified with V. Accordingly, S?P := SC\q-\^P) and .2¾ := Jz?| -i(g) are
viewed as invertible sheaves on F. We then say that S£P and 2q are algebraically
equivalent and write JS?p « .5¾. We can define an equivalence relation on the set
of invertible sheaves on V which is generated by all the relations « as above with
C,S?,P,Q ranging over all possible ones. We denote this equivalence relation by the
same symbol «. Note that if C = P1. then SfP ~ .5¾ (linear equivalence). Hence,
the algebraic equivalence is coarser than the linear equivalence (Problem III.9.4).
Lemma 9.13. Let V be an algebraic surface., and let D. D', E be divisors on V. If
DmD' {i.e., &V{D) « ffv{D')), then (D ■ E) = (£>' • E).
Proof. Since the intersection number ( • ) is a bilinear form, we may assume
that E is an irreducible divisor. The condition D ~ D' means, by definition, the
following: There exist nonsingular projective algebraic curves C\,..., CN, invertible
sheaves S?^ on V x C,- for each ;', and closed points P,. g, on C, for each i such
that
N N
/=1 /=i
where we denote the tensor product (g> for invertible sheaves by the sum+ by
confusing invertible sheaves with Weil divisors. Hence, in view of the bilinearity of the
intersection form, we have only to prove the lemma in the case where &v{D) = Sfp
and &V{P>') = -2¾ for a nonsingular projective algebraic curve C, an invertible sheaf
J? on V x C, and closed points P, Q of C.
Let a:£xC^FxCbethe natural closed immersion. In the proof of Theorem
8.2 (1), we have shown that {SeP • E) = deg(5>|£) and (2Q ■ E) = degCS?e|£), where
deg(5>|£) = deg(a*^|£x(P)) and deg(^e|£) = deg(a*^|£x(e)) evidently. On the
other hand, if v: E —> E is the normalization morphism of E and /? = a • (v x lc),
we have deg(^|£) = deg(f*2\Ex{p)) and deg(Jz?e|£) = deg(^*^|£X(e)). Thus,
we have only to prove the following assertion:
Let E, C be nonsingular projective algebraic curves, let D be an irreducible
divisor on W := E x C, and let P, Q be closed points of C. Then the equality
(D ■ EP) = (D • EQ) holds on W, where EP = E x (P) and EQ = E x (g).
Consider the case where q{D) = C for the natural projection q: E x C —>
C. Let f = q\D. Then (D ■ EP) = dim f%@D ®0C k{P). (In fact, write D n
EP = {P],... ,Rm}. Let t be a local parameter of C at P. At each point P,, t
is taken as one of local parameters of W. Let x, be another local parameter of
W at Rj, and let /,-(*,*,•) = 0 be the local defining equation of D at P,. Then
(D ■ EP) = J2™=\ ^m^ @w.r, /(/i. 0 and the right side of this equality coincides with
dim/*^D ®gc k(P).) Meanwhile, f^D is flat as a coherent ^c-Module. Hence, it
is a locally free &c-Module. (In fact, a torsion-free, finitely generated module over
a principal ideal domain is a free module.) Hence, its rank is constant at various
points of C. Namely, (D • EP) = (D • Eq). Consider next the case where q(D) is a
closed point P of C. Then D = EP. Since g ^ P, {D • EQ) = deg{<?w{D)\Ee) =
deg(^|£e) = 0. We have {D . EP) = {EPf = degJfEp/lv. Since {JtrEp,w)* =
^ep/(^ep)2 = t&wl^^w - &ep, we have (D • EP) = 0. In both cases, we have,
therefore, the equality (D ■ EP) = (D ■ Eq). D

188
III. ALGEBRAIC SURFACES
Let p: V —> B be a fibration. Two fibers V^ ■ F/,, of/? are algebraically equivalent
as divisors on V. (Let T:- \m(\v.p){c V x B) be the graph of p and let J? :=<f (T).
Then Sf\Vxibl) = ^v{Vhl) and 5C\Vxih2) = @v(Vhl)\ Let S be an irreducible curve
on V, and let F be a general fiber of p. Then the intersection number n : = (S ■ F)
is independent of the choice of F. If n > 0. then we call S an n-section of p. In
particular, if n = 1, we call S a section (or a cross-section) of p. If n = 0, either S
does not meet F, or S coincides with F. Since F is a general fiber, we may assume
that S does not coincide with F. Then p(S)isa closed point of B, and S is contained
in a fiber of p. Let F\ be a fiber of p different from F. Then (F2) = (77 • F\) = 0.
If we write F\ = 52/=i «/C,- (a,- > 0. C,: irreducible component), (77] • C,) = 0. By
Lemma 9.3, F\ is connected. Hence, if F\ is a reducible fiber, (F\ - OjCj • C,) > 0.
So (C2) < 0. Extending this argument, we obtain the following result.
Lemma 9.14. Let p: V —> B be a fibration. and let Vn be the generic fiber. Then
the following assertions hold:
(1) (i) Giving a section S of p is equivalent to giving a K-rationalpoint of Vn.'
where K = k{B).
(ii) A cross section S of p is a nonsingular algebraic curve and is isomorphic
to B by p.
(iii) If p has a section, then p has no multiple fibers.
(2) Let Vb = 52/=i aiC' be a reducible fiber of p, and let M = ((C,- • C;))i<;./<r
be the intersection matrix of the irreducible components of Vb. Then M is negative
semidefinite. Namely, for Z = 52/=i ^/0(^/ 6 R). we have (Z2) < 0, and (Z2) = 0 if
and only if Z = aVh (a e R).
Proof. (1) (i) and (ii). Let S be a section of p. Let P be any closed point of
S. and let b = p(P). Let t be a local parameter of B at b, and let y = 0 be a
local defining equation of S at P. Since 1 = (S ■ Vb) = dim.kd?v.p/{t,y), (t,y) is
a system of local parameters of V at P. Hence, &SP = @v.p/{y) and ms.p = (0-
So S is nonsingular at P. Since P is arbitrary, S is a nonsingular algebraic curve
and ps '■= p\s is an unramified morphism. ps: S —> B is a finite morphism and
apparently a one-to-one mapping. (In fact, let B be the normalization of B in k(S).
Then ps factors as
ps-.S^B^B.
and a is a birational morphism between nonsingular algebraic curves. Hence, a is
an isomorphism, and v is a finite morphism as the normalization morphism.) Thus,
&s.p is a finitely generated &B,h-module, and &s.p = @B.b + ttfs.p- So &s.p = @B.b
by Nakayama's lemma. Namely, ps is an isomorphism. In particular, S gives rise
to a /^-rational point. (In fact, the closed immersion S «-► V over B gives rise to a
/^-rational point S xB Spec(T^) «-► Vn by taking fiber products with SpecK —> B.)
Conversely, let £ be a /^-rational point of Vn, and let S = {£} be the closure in V.
Since tr.deg^ A:(£) = tr.deg^/r = 1, S is a complete algebraic curve over k. Since
/>s :— p\s- S —* B is a birational morphism, /¾ is an isomorphism. (In fact, for
P e S and b = p(P), &s,p > &s.b and @B.b is a discrete valuation ring of K. Hence,
@s.p ~ @B.b-) Moreover, with the above notation, let y = 0 be a local defining
equation of S at P. Since @v.pj{y) = &B.b, we have mv_P = (t,y) with a local
parameter t of B at b. Hence, {Vh ■ S) = 1. Namely, S is a section of/?.
(ii) If Vh is a multiple fiber, we can write Vi, — aT with a > 1. If 51 is a section
of/?, we have (F/, • S) = a(T ■ S) > a > 1 which is a contradiction.

9. PENCILS OF CURVES
189
(2) For Z = 52/=i V'C<- we make the computation
(z2)= J2 s's.i(c--cj)
l<i.j<r
E
1</,/<i
— ' —{ajCj -cijCj)
a; Clj
(aiCi+---+arCr-[—) fliC,+...+ ( — ) ar(Cr)
a\ I \a,
m-w «««■■<>
•<j
where («i Q H h o,C • C,) = (Vh ■ C,-) = 0 for each i. We have
(Z2) = 0 if and only if ^ = ^
whenever i ^ j and (C,- • C;) > 0. Since V/, is connected. .s-,/a,- is a real number
independent of i. So (Z2) = 0 if and only if Z = a Vh for some a e R. □
III.9. Problems
1. Let A: be a field of characteristic zero, and let A be a ^-algebra. We assume
moreover that A is an artinian local ring and a finitely generated ^-module. Then,
following the subsequent steps, verify the assertion that A is reduced if and only
if <V = (o).
(i) Show that 0.A/k — (0) if -^ is reduced.
We prove below the converse.
(ii) For a maximal ideal m of A, mN = (0) for some N > 0. Let A = A/m2.
Then Q^,k = (0) if Q^ = (0). Hence, replacing A by A. we may assume that
m2 = (0).
(iii) The residue field k' :— /1/m is a separable algebraic extension of k. So
let A: be a normal extension of k containing k'. and let A = A (¾ k. Then A
decomposes as A = A\ x • • ■ x A,- (a direct product), and A, is an artinian local
ring with residue field k. If each A-, is reduced, then A is reduced as well.
(iv) We may replace A by /4,-. and hence, by doing so. assume that k' = k. Then
&A/k '^>a k = m. Hence. Q^. = (0) implies m = (0).
2. Let X be a reduced algebraic scheme defined over an algebraically closed field k.
If @x.x is a normal ring at every closed point x of X. show that each connected
component of X is irreducible.
3. Let A be a linear pencil on an algebraic surface V. and let D\.Di be distinct
two members of A. Show that BsA = SuppZ)t n SuppZ)2.
4. Let V be an algebraic surface, let C = P1. let 21 be an invertible sheaf on V x C
and let P. Q e C. Following the subsequent steps, prove that SfP := 2\Vx(P)
and .2¾ := .S?|Fx(G) are linearly equivalent.
(i) We can take an inhomogeneous coordinate t on C so that P. Q are defined
by t = 0. t = 1, respectively. So we may assume below that C = A1.
(ii) For x 6 V.@Vx[t] is a UFD. So the pullback of 2 onto Spec^Kv[/] is a

190
III. ALGEBRAIC SURFACES
trivial invertible sheaf. Hence, there exists an open neighbourhood U of x such
(iii) There exists an open covering {£/,},G/ of V such that J?|c/,xc — &u,xc-
Show that, for the transition functions {/,7)/.,6/ of Ji? with respect to the open
covering {£/,■ x C}i€/ we have /,•; g r(Ujj.(fv)*. where £/,-, = £/,- n Uj. Hence,
there exists an invertible sheafs on V such that 5f = q*Jf, where q: V xC —> C
is the natural projection.
(iv) Show that SeP = SCq.
5. (v) Let D.D' be divisors on V such that D' = D + (/) for / g fc(F). Let
(/?: V —> P1 be the rational mapping, x 1—> (1. /(x)). defined by /. and let T be
the graph Im( 1,/ x ip){Q V xP1) of tp. Let ^ = <?(r) <8><?(Z) x P1). Show that
-^(00) = ^(-° + -4) ancl -^(0) = ^>(-D' + A)- where ^ is some divisor.

CHAPTER 10
The Riemann-Roch Theorem
for Algebraic Surfaces
In Chapter 6 of Part II. we defined the canonical Module mX/k and the canonical
divisor Kx for an w-dimensional. nonsingular algebraic variety X defined over k.
For the generic point n of X and K = k(X). we have mx/k.n = A" &K/k- Here if
{xi x„} is a system of local parameters at a closed point x of X. we can write
A" O-K/k = K dx\ A ■ • • A dx„. So if cw is a nonzero element of A" ^K/k- we can
write m = fP dx\ A • • • A dx„ with fP £ K. We consider the divisor (fP) on a
coordinate neighbourhood UP of the system of parameters {x\ x,,}. If P moves
on V. then the divisors (fp) are patched together to give a divisor (cw) which is
linearly equivalent to Kx-
In what follows let V be an algebraic surface. All algebraic curves on V are
supposed to be closed subschemes unless otherwise mentioned.
Lemma 10.1. Let C be an irreducible algebraic curve on V, and let JV be the
defining ideal sheaf of C. Then the following assertions hold:
(1) The following sequence is an exact sequence
o - fclfl - "i-aIc - nc/k - o.
(2) IfC is a nonsingular algebraic curve, then we have mc/k — (cojy/< ®^c) ®-^c/r-
If g(C) is the genus of C. we have
g(C)=l-(CC+Kv)+\.
Furthermore, we have the following equalities:
X(&v(-C))=l-(C.C+Kv)+X(&y).
X(@v(C)) = \{C-C -Ky)+X(<?y).
Proof. (1) By (II.6.18), the following sequence is exact.
Sc/s£-^toy/k\c^nc/k^o.
where Qj,-/a-|c is a locally free &c-Module of rank 2. For a point Q e C, C is
defined by an equation / = 0 locally at Q. Then (J'c/J'Dq = @c.Qf with f = f
(mod(jrce)2)- Hence. Jcj^c is an invertible sheaf. Put & = @c.q- @ is an integral
domain, and (D.v^\c)q — & ®@ has no torsion elements. So, aQ: (J^c/^Dq —>
(Qy/k\c)Q is zero or injective. If ae were zero, then (Oc//v.)g = & © & which is a
191

192
III. ALGEBRAIC SURFACES
contradiction. Thus, ag is injective. Since Q is an arbitrary closed point, a itself is
injective.
(2) In the exact sequence of (1). Qc/k is also an invertible sheaf. Hence.
v/k\c - Qc/a- ® {^cI^1)- This implies that coc/k = {coV/k\c)@^c/v- On the
other hand. deg(cor/k\c) = {Kv ■ C) and deg-/fr/r = -deg(<fr(-C) ®ffv @c) =
(C2). So we have
2g - 2 = deg»c/, = (Kr ■ C) + (C2).
This gives rise to the equality concerning g. In order to obtain the remaining two
equalities, we make use of the following exact sequence
(*) o-><M-c) -+®v -+@c -+o.
Taking the Euler-Poincare characteristics, we obtain x(^v(—C)) = xi@v) - xi@c)-
Since xi&c) = 1 -hx{C.@c) = 1 - g(C). we obtain the equality
X(0A-C)) = ^CC+Kr)+x(0v)-
Next, applying ®$,-(C) to the exact sequence (*). we get an exact sequence
0-^,,-^ &V{C) -► ffc{C) -► 0.
where ^c(C) := &c ®<s\ @v{C). Making use of the Riemann-Roch theorem on C.
we obtain the equality:
x(&v(C))=x(&v)+x(0c(C))
= X(^) + (C2)-l-(CC + Ky)
= X-{C-C-Kv)+x{0v). □
Definition 10.2. For a divisor D on V. we define the arithmetic genus pa{D) by
Pa(D)=l-(D-D+Kr) + \.
We will sometimes refer to this equality by calling it the arithmetic genus formula.
We shall show in Lemma 10.4 that pcl(C) > 0 if C is an irreducible divisor.
Lemma 10.3. Let P be a closed point of V. let a: V' —» V be the monoidal
transformation with center at P. and let E = a~x(P). Then the following assertions
hold:
(1) Kv ~ a*(Kr) + E and {Kv,)2 = {Kr)2 - 1.
(2) Let JFp be the defining ideal sheaf of P on V. For every nonnegative integer
n. we have
o*@v\-nE) = SnP. Rxo*@v,{-nE) = (0).
where J^ = &v.
(3) Let C be an irreducible divisor on V, let p be the multiplicity of C at P, and
let C = (t'(C). Then we have
Pa(C) = Pa(C) - l-p{ju - 1). x&c) = X(<?c) + \fiili ~ I)-
Proof. (1) Choosing a system of local parameters (u. v) at P. set co = du Adv.
w = v/u is an inhomogeneous coordinate on E. and co = udu A dw if we consider

10. RIEMANN-ROCH FOR SURFACES
193
(u.w) instead of (u.v). Considering the divisors {co)y and {co)y which a nonzero
element co of /\" Qjr/yt defines on V and V. respectively, we have coy, = a* (co) v + E.
Hence. Kyi ~a*Ky + E. By virtue of Lemma 8.8 and Corollary 8.9. we get {Ky)2 =
{Kyf-\.
(2) Set & = &y,P. m = myp. W = V x v Specif and t = a x v Spec^.
We have only to show that x„&w(-nE) = m" and Rxr„cfw(-nE) = (0) for each
n > 0. It is easy to see that a%@y = &y. Hence. r*cfw = cfv. Noting that
xti'&w =@w{-nE). wegetT^H/(-«£) = r*(m"$V) = m"rtcfw =m". We shall next
show that R}x*{m?@w) = m"RXTtcfw. Since m" is a finitely generated ^-module, we
write m" = ^, = 1 f\@. So we obtain a surjection
N
@fi<9w^m"ffw-*Q.
/=i
Noting that R2t*9 = (0) for any coherent ^-Module 9. we obtain from the
surjective homomorphism ft the following surjection:
N
@Rxx*{fi0w) -* Rlu(m"&w) -+ 0.
/=i
where Rxr^(ficfw) = fiRxxtt(@w). (Check this directly from the definition.) Hence,
it follows that Rxxif(van&w) = m"Rxzt(cfw). In the cohomology exact sequence
(concerning R'z*(-)) associated with an exact sequence
0 -» ITU?w -» @W ~+ @E ~+ 0.
we have RxztcfE = Hx(E.cfE) = (0). Hence, the cohomology sequence yields
mRxz*{ffw) = Rxzt(cfw). Since r is a proper morphism. Rxt*(®w) is a finitely-
generated ^-module (compare (11.4.18)). So we get Rxt*(@w) = (0) by Nakayama's
lemma. Hence, we obtain Rxx)ftfw(-nE) = m!1RxT*tfw = (0) for each n > 0.
(3) By Lemma 8.8, c*(C) = C +fiE. By means of the arithmetic genus formula,
we have
Pa(C')=l-(C'.C'+Ky) + l
= 1-(<J*(C) -fiE ■ <j*(C+Ky) -(ji- \)E) + 1
= Pu{C) - -nip-- 1).
We shall verify that the second equality holds. We have x(&c) =x(.$y)—x($v (-C))
&ndx{<?c)=x(0v)-x(M-C))- Furthermore, X(0y) = X(0y) md X{0y,(-C -
fiE)) = x(@v{-C))- (In fact, for a quasicoherent ^'-Module 9 and an invertible
sheaf 2 on V, Ria*{a*2 <g> 9) = 2 <g> R^a^) for each q > 0 (problem 1.2.9).
We have, moreover, a spectral sequence (1.2.28)
em = hp{V.Rl,a%(a*2 ® 9)) =*> Hp+<<{ V. a* 2 ® 9).
Apply these two results to {2.9) = (tfy.tfy) and {cfv{-C).cfy,). In case {3.9) =
[cfy.cfy). by Lemma 10.3 (2). o*@y = &v and Ria^y = (0) for each q > 0.

194
111. ALGEBRAIC SURFACES
Hence. H''(V.^r) = HP{V'.0v,) for each/? > 0. Thus, xi^v) = xi&v)- In case
{SCP) = (@V(-C).@V,). we have
R«o*{0v{-C'-nE)) = ]?a.{a*<fv{-C)) = ffy(-C) ® R*a*{ffy).
Hence, we obtain HP{V.ffv{-C)) = W{V.<9V,(-C - fiE)) for each p > 0. So.
x{@v{-C - fiE)) = xX@v{-C)).) Meanwhile, for 1 < i < ju. we have an exact
sequence
0 -► &V\-C -uE) -+ &V\-C - (i - \)E) -+ @E{-fi + »' - 1) -► 0.
From this we obtain
X(@v<(-C -ME)) = x(*i"{-C' - (/ - \)E)) -x(&f.(-M + ' - 1))
= x(0r,(-C>-(i-l)E)) + (M-i).
So X\@v{-C' -fiE)) = xi^i"{-c')) + jm(m ~ 1)- These observations yield the
required equality
x{0c.) =x{*c) + ^-1)- n
With the notations of Lemma 10.3. each point on E is called an infinitely near
point of the first order of P. When C is an irreducible algebraic curve passing through
P. the intersection points of £ with the proper transform C of C are called infinitely
near points of the first order of P lying on C. If C is nonsingular at P. we have
fi = 1 = (C ■ E). So C meets £ in a single point, and C and E cross normally at
this point. However, if C is singular at P. C might meet E in more than one point
and C might be singular at an intersection point of C and E. When P'(e C n E)
is a singular point of C. the multiplicity of C at P' is called the multiplicity of C at
P'. Let t : K" -+ F' be the monoidal transformation with center at P' on E, and let
£' = t_1(P'). A point P" on E' is an infinitely near point of the first order of P'\
we call P" an infinitely near point of the second order of P. When we speak of P"
lying on the algebraic curve C. we signify that P" is one of the intersection points
of the proper transform C" = ((tz)'C with E'. In a similar fashion, we define an
infinitely near point of the «th order of P. which we call simply an infinitely near
point without caring about its order n.
Lemma 10.4. Let C be an irreducible algebraic curve on V. and let C be the
normalization of C. Then we have
(\)Pa(C)>0.^
(2) pa{C) ~ g{C)+^2P jfip{fip— 1). where P ranges over all points of C including
infinitely near points and ftp is the multiplicity of C at P.
Proof. Let v. C -+ C be the normalization morphism. Consider the exact
sequence
o -+ &c -+ Vit@d -+ m -+ o.
where 3t = v^&^l&c ■ By virtue of the observation made just before Theorem 7.3.
we have x(&c) < l{@c)- With C as in Lemma 10.3. v factors as
where /? = a\c> and a is a birational mapping defined by a := /?_1 v which becomes

10. RIEMANN-ROCH FOR SURFACES
195
a morphism by (II.4.29). Clearly, a and /? are finite morphisms. Considering exact
sequences for a and /? similar to the above one for v. we obtain x($c) < x(&c) <
x{@c)- Note tnat #(^c) -xi@c) = \ju(m ~ 1)- This implies that C has only finitely
many points P with multiplicity /iP > 1 including infinitely near points. Hence,
repeating the monoidal transformations with centers at points of C finitely many
times, we obtain a nonsingular algebraic curve which is birational to C. It is obvious
that this nonsingular curve is isomorphic to C. By Lemma 10.3. pa(C) = g(C) +
^2p \npijxp - 1). Since g{C) > 0, pa{C) > 0 follows. □
Lemma 10.5. Let C be an irreducible algebraic curve on V. Then teh following
assertions hold:
(l)hl(C.0c)=Pa(C).
(2)x(0v(C))=l1(CC-Ky)+x(&v).
Proof. (1) With the notations of Lemma 10.3. x(&c) + pa{C) = xi&c) +
pa(C). The proof of Lemma 10.4 implies also that there exists a birational morphism
t: W —> V. which is a composite of monoidal transformations, such that the proper
transform of C by t is isomorphic to the normalization C of C Therefore, we have
X(0C) +Pa(C) = x{<9d) +Pa(C) = 1 - h\C.^)+g(C) = 1.
This yields hl(C.@c) = P«{C).
(2) The equality xi@v{C)) — x($v) + x(&c(C)) follows from the exact sequence
Q^(9V -^<fv{C) -*&C(C) ->0.
Applying the Riemann Roch theorem (III.7.3) on C. we have
x(&c(C)) = (C2)+X(0c)
= (C2) + l-Pa(C)
= (C2) + i-l(c.C+Ky)-l = ±(C-C- Ky).
This yields the equality (2). □
We shall next state the Riemann-Roch theorem on an algebraic surface which is
the purpose of the present chapter.
Theorem 10.6 (Riemann-Roch Theorem). Let V be an algebraic surface, and
let D be a divisor on V. Then the following assertions hold:
(1)X(MD)) = {(D.D-KP)+X(0V).
(2) ti{D) = h2-'(Kv-D) (i =0.1.2). where h'(D) = h'{V.@y{D)).
Proof. (1) Case D > 0. Writing D = ^, a, Q, (a, > 0. C,: irreducible), we
prove the assertion by induction on £\ a,-. If D is an irreducible divisor, then the
assertion is proved in Lemma 10.5. When we decompose D = A + C (A > 0. C:

196
111. ALGEBRAIC SURFACES
irreducible component), we shall prove the assertion, assuming that the equality (1)
holds for A. From an exact sequence
0 -► &V{A) -* &V{A + C) -^&C(A + C) -► 0.
we obtain by virtue of Theorem 7.3 and Lemma 10.5
X{MA + C)) = x(0r(A))+X(0c(A + O)
= 1-{A-A- Kv) +X{0r) + {C-A + C)-^{C-C+KV)
= -{A + C-A + C-Kr)+x(*r)
= l-(D.D-Kr)+x(0v).
In case D is not an effective divisor, write D — A — B with A > 0 and B =
S; bjCj > 0. We shall verify equality (1) by induction on £\ bj. When ][\ bj = 0.
the equality (1) has been established as above. So decompose D = A - (F + C)
{F > 0. C: irreducible component) and assume that (1) holds for A - F. We shall
show that (1) holds for D. Making use of the exact sequence
0 -► &y(D) -► &V{A -F)^@C{A-F)^ 0.
we compute as follows:
X(MD)) = X(MA - F)) - X(&c(A - F))
= X-{A-F-A-F-Ky)+X{@v)-(C-A-F)
+ l-(CC+Ky)
= X-{A-F-C-A-F-C- Kv) +x(*r)
= ^(D-D-Ky)+xMv)-
(2) It follows from (II.6.29). □
For a positive integer n, we call P„ := h°(V.&y(nKy)) the n-genus of V. We
denote P\ by pg and call it the geometric genus of V.
Lemma 10.7. Let z: W —> V be a birational morphism of algebraic surfaces which
is a composite of monoidal transformations with centers at points. Then we have
(1) h°{W.@w{nKw)) = ti)(V.@y(nKv))Jor each n > 0.
(2) hx(W.@w) =h\V.0v).
Proof. (1) It suffices to prove the assertion in the case where z is a monoidal
transformation with center P. Let E = r~l(P). Since Kw ~ z*Kv + E. we have
H\W.<9w(nKw)) = H\V.z*{z*&>v{nKv) ®&>w{nE)))
= H°(V.0v(nKv)®T.0w(nE)).
Here it suffices to show that z*@w{nE) = &y. Let U be an open neighborhood of
P in V. Then H°{U.z*&w{nE)) = H°{z-l{U).cfw{nE)). If we regard &w{nE)
as contained in the function field k{W) = k(V). f G H°{z-l{U).cfw{nE)) is a

10. RIEMANN-ROCH FOR SURFACES
197
rational function on W which, restricted to r-'(£/). has poles along E of order at
most n. Since t~x{U) - E ^ U - {P}. if we consider f <G k(V), f is a rational
function on V which is regular on U except for the point P. Since U is nonsingular,
f e H^iU.&u). Hence, f e H*)(t-x(U).@w). So r^w{nE) = &v.
(2) As in (1) above, it suffices to prove the assertion in the case where t is a
monoidal transformation with center at P. The spectral sequence
£/>■</ = Hp{V.Riu<9w) => Hp+i{W.0w)
yields an exact sequence (1.2.26).
0^Hl(V.r„&w) -^Hl(W.&w) -^H°(V.R1t^w).
By Lemma 10.3. %%&w = &v and Rxxit&w = (0). So we obtain H\V.@V) ^
H\W.@W). D
As proved in the next chapter, a birational morphism x: W -+ K of algebraic
surfaces is necessarily a composite of monoidal transformations with centers at points.
Hence. Pn. pg := h®{V.t?v{Kv)). and g(K) ;= hx(V.(9v) are quantities common to
algebraic surfaces V with the same function field. A quantity possessing a property
is called a birational invariant. We call q{V) the irregularity of an algebraic surface
F.
As an application of the Riemann-Roch theorem, we shall prove the Hodge index
theorem. For this, we need some preparation.
Definition 10.8. Given a divisor D on an algebraic surface V. we say that D
is numerically equivalent to zero if (D ■ C) = 0 for every irreducible algebraic curve
C on V; D = 0 by notation. Two divisors are numerically equivalent. D\ = Di by
notation, if Dx - D2 = 0. Put Div°(F) := {D e Div(F);£> = 0}. Div°(K) is a
subgroup of Div(F). Put NS(F) := Div(F)/Div°(F), which we call the Neron-
Severi group. It is known that NS( V) is a finitely generated abelian group. Hence.
NS(F)R := NS(F) ®z R is a finitely generated R-module. whose rank we call the
Picardnumber and denote by p( V). The intersection number (•) for divisors induces
a nondegenerate quadratic form on NS(F)R which we call the intersection form.
Here the intersection form being nondegenerate signifies that {D ■ G) = 0 for each
G e Div(F) implies that D = 0.
By Lemma 9.13 and Problem III.9.4. there are the following implications among
the equivalence relations on divisors:
D\ ~ D2 (linear equivalence) => D\ w D2 ( algebraic equivalence)
=> D\ = Z>2 (numerical equivalence).
The Hodge index theorem, which we state below, asserts that the signature of
the intersection form on NS(F)r is (l.p(V) - 1).
Theorem 10.9. Let V be an algebraic surface, and let A be a divisor on V with
{A1) > 0. For a divisor D. if {A • D) = 0. then either {D2) < 0 or D = 0.
Proof. Proof consists of three steps.
(1) We shall first prove that the stated assertion is equivalent to the assertion
that the signature of the intersection form is {l.p(V) - 1). Let [A] be the numerical
equivalence class of A. [A] ^ 0 by the hypothesis. Let M be the orthogonal subspace
to [A] in NS(F)r. If the signature is (l.p(V) - 1), then the intersection form

198
III. ALGEBRAIC SURFACES
restricted on M is negative definite. (Here we use Sylvester's law of inertia on real
quadratic forms.) The numerical equivalence class [D] of D is an element of M.
Hence, if D ^0, then we have (D2) < 0. Conversely, if the stated assertion holds
true, the intersection form restricted on M is negative definite. So the signature is
(l.p(V)-l).
Let H be an ample divisor on V. By Sylvester's law of inertia, we have only to
prove the theorem when A = H. We assume below that A = H.
(2) We shall show that the assumption (D ■ H) = 0 and {D2) > 0 leads to
a contradiction. For an integer n. we apply the Riemann-Roch theorem to nD to
obtain
x{0v{nD)) = X-{nD • nD - Kv) +X{0y).
Since hx{V.@v{nD)) > 0. the above equality gives rise to the following inequality:
dim|w£>|+dim|A:,'-n£>| > -»2(Z>2) - -n{D ■ Kv) + xi&v) ~ 2.
Since (D2) > 0 by the hypothesis, the right side of the equality —> +oo as \n\ —> oo.
Hence, dim|»Z)| + dim|AT[/ - nD\ —> +oo as |n| —> oo.
If dim|«Z)| —> oo and dim \KV —nD\ —> oo. then dim |Kv\ > dim|AT|/ —nD\ —» oo
as \n| ^oo. (In fact, take D0 e\nD\. ThenF + Z>0 £ \KV\ for each F e \KV -nD\.
Namely. Z>0 + \KV - nD\ C \KV\. and we may consider H°{V.@V{KV - nD)) C
H°{V.@y(Kv)).) Since dim|ATr| < pg - 1. this is impossible. Suppose dim|ATK -
nD\ —> oo as n —>■ +oo and as n —» -oo. Then taking E„ e \Ky — nD\ (n > 0),
dim\2Kv\>dim{En + \Kv+nD\) > dim \Kv+nD\. So dim \2KV\ -» oo as n -> +oo.
This is a contradiction. Hence. dim|«Z)| —> oo as « —> +oo or as « —» —oo. If
dim \nD\ —> oo as « —» +oo. then {D ■ H) > 0 (Lemma 8.6). If dim | - nD\ —» oo as
n —> +oo. then (D • H) < 0. In either case, we get a contradiction to the hypothesis
{D ■ H) = 0. Hence, we must have {D2) < 0.
(3) We shall show that D = 0 if (Z>2) = 0. Suppose D £ 0. Then (£> • £) ^ 0
for some divisor E. Replacing E by £' := {H2)E — (£ • H)H, we may assume that
(E ■ H) = 0. Set D' :=mD + E. Then (£>' -//)=0 and (D'2) = 2m(D • E) + (E2).
Hence, we may assume CD'2) > 0 by choosing a suitable integer m. Applying the
argument in (2) to D'. we have (D' ■ H) ^ 0. This is a contradiction. So D = 0 if
(/)2) = 0. □
111.10. Problems
1. Suppose the characteristic of k is zero. Let C be an algebraic curve of degree n
on P2 defined by the following equation:
Y2Z"-2 = (X - axZ) ■ ■ ■ (X - a„Z).
n>3. a,- e k. a,- ^ aj. (i ^- j).
Verify the following assertions:
(1) C is irreducible.
(2) If n = 3. then C is nonsingular: if n > 3. then the singular points of C are
Poo := (0.1.0) and its infinitely near points P\£ (1 < i < [n/2] - 1).
(3) pa{C) = \{n - \){n - 2) and g{C) = n - [n/2] - 1. where C is the
normalization of C.
2. Let C be an irreducible algebraic curve on an algebraic surface V. Prove that
the following four conditions on C are equivalent to each other:

10. RIEMANN-ROCH FOR SURFACES
199
(1) (C2) <0and (Kv ■ C) < 0.
(2) (C2) = (tfr.C) = -l.
(3) (C2) = -1./>,((C)=0.
(4) (c2) = -1. C =P'.
An algebraic curve C satisfying these conditions is called a (-1) curve.

CHAPTER 11
Minimal Algebraic Surfaces
Let V0 be an algebraic surface, let a: V -+ Vq be the monoidal transformation
with center at P of Vo and let E = (J~X{P) be the exceptional curve. Then E = P1
and (£2) = -1 (Lemma 8.8). Namely, £ is a (-1) curve on V (Problem III.10.2).
The converse of this result holds.
Theorem 11.1 (Contractability criterion of Castelnuovo). Let V he an algebraic
surface, and let E be a (—1) curve on V. Then there exist a {nonsingular} algebraic
surface, a birational morphism a: V -+ Vo. and a point P of Vo such that a: V —
E A Vo — {P} and a{E) — P. a is the monoidal transformation with center at P. We
call a the contraction of the ( — 1) curve E.
Proof. Proof consists of four steps.
(1) Let H be a very ample divisor on V such that H\V.&{H)) = (0). To find
such H. we take any very ample divisor H on V. and replace H by nH{n >• 0)
if necessary (II.5.13). Set a := {E • H). We may assume a > 2. Then we have
H\V.@{H + iE)) = (0) for each i, 0 < / <a.
We shall prove this result by induction on i. If i = 0. the result holds by the
choice of H. Assuming that the result holds for i — 1. we shall prove the result for
i. From the following exact sequence
o -► 0{H + (i - \)E) -► &{H + iE) -> <9E(a - i) -> 0.
we obtain the cohomology exact sequence
HX(V.@{H + (i - \)E)) -► H\V.0{H + iE)) -► Hx{E.@E{a - i)).
where the left and right terms are zero. So HX{V.@{H + iE)) = (0).
(2) We shall prove that Bs\H + aE\ = 0 and V := <l>\H+aE]: V -+ PN {N : =
dim \H + aE\) induces an isomorphism V — E A V\ - {Pi}, where V\ := f{V) and
Pi := f{E) is a point of V\.
Since H is a very ample divisor. \H + aE\ has no base points outside E. Hence,
if we show that \H +aE\ has no base points on E. we will see that Bs\H + aE\ = 0.
For this, look at an exact sequence
0 -> &{H + {a - l)E) -+ &{H + aE)^(9E^G
and the associated cohomology exact sequence
0 -► H°{V.^{H + {a- \)E)) -+ H°{V.0(H + aE)) -^ H°(E.0E)
-+ Hl(V0(H + (a - l)E)).
By (1). Hl{V.@{H + (a - 1)£)) = (0). Hence, the homomorphism p is surjective.
201

202
111. ALGEBRAIC SURFACES
Since p is induced by the restriction of &v(H + aE) onto E. if p is surjective there
exists D e \H +aE\ such that £nSuppZ> = 0. So \H + aE\ has no base points on E.
Thus. Bs\H + aE\ = 0. In particular, ip := <S>\h+ue\ isamorphism. Let M = dim |/f |.
Since \H\ + aE C \H + aE\. we obtain $>\H\+aE '■ V —» PM as a composite of ip and
the projection from P^ onto PM. As <S>\H\+aE gives an isomorphism between V — E
and its image. y> also gives rise to an isomorphism between V — E and its image.
f{E) is apparently a point P\ because E C SuppD or £ n SuppD = 0 for each
D e \H + aE\.
(3) Let V0 be the normalization of V\ in the function field k{V). and let v: V0 -*
V\ be the normalization morphism. Since V is a normal algebraic variety, the mor-
phism ip: V —> Fi factors as
^: F^ F0^ F,.
where <r: F —» F> is a birational morphism and a{E) is a point P of Fo. Furthermore.
<7*$V = $V0- (In fact, since a^&v is a coherent $V0-Module. F0 would not be a normal
algebraic variety if &v^ C a^&v) We shall show below that P0 is a nonsingular point
of F0.
Put A = ffva.p and m = mro/>. Restricting the morphism a: V —> F0 to a
neighborhood of P0. we shall discuss with <r x,/0 Spec .4: F x ,¾ Spec .4 —» Spec.4.
Let J^ = J^- be the defining ideal sheaf of £ on F. and let £„ be a closed subscheme
defined by <yn+1. So £o = £, and £„ has the same topological space as £ though
the structure sheaf of £„ is enlarged by the nilpotent elements. Since a~x (P) = £ as
sets. ,J = y/m&y. Hence J?N c nvfy C J^ for some N > 0. If r > ». £„ is a closed
subscheme of £r. So there is a ring homomorphism 0„r: //°(£,-. (¾.) —>H°{E„.&Eii).
and we can thus consider the projective limit lim H°{E„.<$EJ. The results to be
< 77
proved are summarized in the following lemma. □
Lemma 11.1.1. (i) 9„, is surjective if r >n.
(^ H°(E„.0Eji) = k[[x.y]]/(x.y)"+K
(iii) A=\im H°{E„.&Eii).
< n
Proof. (1) Since £ = P1 and (£2) = -1. we know that
(*) sn+l/sn+2 = {j?/.y)®("+V) = <?Pi(« + 1) (n > 0).
Thus, we obtain an exact sequence
0 -► ^P, (» + 1) -► <f£„+l -> ^£„ -► 0.
Since //'(P'.^p^n + 1)) = (0) (n + 1 > 0). we obtain the next exact sequence
(**) o - H°(Pl.fi¥l{n + 1)) - H°(En+[.d?E,i+l) ^ H°{En.<9Eii) - 0.
In particular. 9n.„+\ is surjective. Since 9nr = #„.„+i • 0„+i.„+2 ■ • • #,--1./- if r > n. 0„r
is surjective.
(ii) In the case n = 0. H°{Eo.0Eo) = k and /z°(P'.^P,(1)) = 2 in (**). If we
choose a £-basis {x.j} of JHr°(P1. ^P,(1)). H°{Ei.@El) = k[[x.y]]/(x.y)2 because
H°(Ei.(fEl) D k. By induction, we assume that H°(E„.<fEi) =
k[[x.y]]/(x.y)n+l and prove the result in the case n + 1. Pull back elements x.y
of H°{En.@Eii) to elements x'.j' of H°{En+\.@E„+l) by 0„.„+i. Then by (*). we see
that ff0(P'.<?pi(» + 1)) has a it-basis {x'n+x.x'ny'.... .y'n+l}. So H°{E„+l.#Eii+l) =
k[[x'.y']]/(x'.y')n+1. Then we have only to denote anew x'.y' by x.y.

11. MINIMAL ALGEBRAIC SURFACES
203
(iii) We have SnN C mn@v C S" for each n > 0. So we have a decomposition
0n-i.n;v-i: H\&vlJnN) -+ H°(0y/m"&y) *+ H\@vlSn).
where p„ is a surjection. We shall show that H°{&v/m"&]>) = A/m" if n 3> 0. First.
look at the following natural exact sequence:
0 _> H°{V,m"0y) -+ H°{V.0,>) -> H°(0y/mn0y)
-^Hx{V.m"@y).
The spectral sequence
£?•» = /f"(Spec^./?«ff*(m',<fi-)) => Hp+"{V.m"&v)
entails //1 (F.m"(fF) =i/°(Spec^. ^'^(m"^)). (By (II.3.4). /T(Spec Act,.(m"<fK))
= (0) (( = 1.2). Next apply (1.2.26).) Meanwhile. Rla*{m"@v) = mnRxa^v
as in the proof of Lemma 10.3. Since Supp/?1^,- C {P}.m"Rla^v = (0) if
n » 0. Hence. Hx{V.mn@y) = (0) (« » 0). Noting that H\V.<9V) = A and
i/°(F,m"(fi/) = m", we see that H°{&y/mn&y) = A/m" [n » 0). By the above
observations, we obtain a surjection p: A —> lim H°(E„.$e„)- where p ~ lim /;„.
Suppose p„N{a (mod m"")) = 0 for a 6 A. Then a (mod mnN) 6 H°{V.Sn") C
H°(V.m"<?v) = m". So a = 0 in A Thus, /> is injective. □
By virtue of the above lemma, A= lim H°{E„.@E„) = lim k[[x. y]]/{x. y)n+x =
< n * n
&[[x.j]], and A is thus a formal power series in two variables. By (II.6.12). A is a
regular local ring. We have thus shown that P is a nonsingular point of Vq.
(4) We shall show that a: V —> Vo is the monoidal transformation with center
at P. Let t: V —> Fo be the monoidal transformation with center at P. Then
y :— t_1 -a: V —> J7' is a birationalmapping, and F-is and J7' -is' (is' := t~'(.P))
are isomorphic to each other by y. & := &v.e is a discrete valuation ring of k(V)
and <f > <f Vo,p. Denote by v the valuation associated with &. As shown in the step
(3) in the proof of A = k[[x, y]], we can choose a regular system of parameters {x.y)
of <fV0./> such that {x, y) gives rise to a &-basis of H°{E.^/^2). Then x/y e & *,
whence v(x) — v(y). Since &vi.e> = {&v0.p\y/x]){.\-)- ® — @v.E'- Since &v>_e> is a
discrete valuation ring of k{V) as well, we have <fV.£ = $v.e>- This shows that the
generic points of E and E' correspond to each other under y. Next let Y c V x V
be the graph of y (compare (II, Chapter 4)), let W be its normalization, and let
q: W —> V be the natural projection. Then q is a birational morphism. We shall
show that q is an isomorphism. Since the birational mapping y (and y ~') between
V and V is isomorphic at the generic points of irreducible curves, q induces also an
isomorphism at the generic points of irreducible curves. Let Q be a ^-rational point
of W. Then &w.q = Dpec @w.c since W is a normal algebraic surface, where C
ranges over all irreducible curves on W passing through Q (compare (1.1.33)). Set
P' = q{Q). Then q(C) ranges over all irreducible curves on V passing through P'.
Namely, &VPi = \~\n^c^v.q(cy Here since &w.c = @y.q(c)- we nave @w.q = @v.p>-
Suppose q{Q) = q{Q') = P' for a ^-rational point Q'. The above observation shows
that &w.q = @w.Q'- So Q = Q'. Thus, q: W —> V is an isomorphism. Similarly,
the natural projection q': W —> V is also an isomorphism. Hence, y: V —> V is
an isomorphism. □

204
III. ALGEBRAIC SURFACES
Corollary 11.2 (Contractibility criterion of Castelnuovo). With the notations
of Theorem 11.1, &VP = {\QeE@v.Q-
Proof. Since a{Q) = P for each QeE.we have <fVQ > @va.P. So f)QeE &v.q 3
@vivp- On the other hand, let C be an irreducible curve through P, and let C" be its
proper transform by a. Since C'n£/|. <fr.c' 3 Hgef @v.Q- Smce ^i'.c = ^V0.c
if C ranges over all irreducible algebraic curves through P. we have
<9y«.p = n *vc = n ^-2 n ^-
Fee Fee ee/r
Thus. ne€£^'.e= ^o.f. □
The following result is called the factorization theorem of hirational morphisms.
Theorem 11.3. Let f: V —> W be a hirational morphism of algebraic surfaces.
Then f is a composite of monoidal transformations with centers at points.
Proof. If dim/ ~l(P) < 0 for each P £ W. then / is an isomorphism by the
same reasoning as in the step (4) of the proof of Theorem 11.1. Suppose / is not
an isomorphism. Namely, suppose dim/~'(P) = 1 for some point P of W. Let
E\ E„ exhaust all irreducible components of / _1 (P). Then the following result
holds.
Lemma 11.3.1. (1) (£,2) < 0 for each i. \<i <n.
(2) The intersection matrix ((is,- ■Ej))\<ij<„ ofE\ E„ is negative semidefinite.
(After the theorem is proved, we shall see that it is negative-definite.)
(3) Kv ~ / *Kw + 2~Z/ ajFjfor ea°h aj > 0. where {Fj} moves over all irreducible
algebraic curves on V whose images by f are points of W.
Proof. (1) and (2). There are only finitely many points of P £ W such that
dim/-'CP)>0. Let {Px P,} exhaust these points. Then/: V-\Jj={ f~\Pi)^
W — {P\ Ps} is an isomorphism (compare the argument in the step (4) of the
proof of Theorem 11.1). Now let H be a very ample divisor on W. Then there
exists an effective divisor H\ such that H\ is linearly equivalent to H and Supp H\ n
{P\ Ps} = 0. Then Hi is identified with / *H\. We may assume that H = H\.
Then (H2) = (/ *H)2 > 0 and [E,- ■ f*H) = 0. Since E, ^ 0 clearly. {E}) < 0 by
Theorem 10.9. Similarly, since (£•_, a,-£,- • / *H) = 0. (^=1 a,-£,-)2 < 0. Hence,
the intersection matrix ((£, • £7))i<,7<„ is negative semidefinite. If we know that
E\ E„ are numerically independent, i.e., a\E\ + ■ ■ ■ + a„En = 0 implies a\ =
■ • • = an = 0, ((Ej • Ej))\<j.j<„ is negative definite. We shall prove this later.
(3) Let {x. y} be a regular system of parameters of W at the point P. Set £ =£,.
and let u be a generator of the maximal ideal of &v.e- Since &v.e > @w.p- we can
write x = u'£,. y = usn. r.s > 0, £,.n £ ($v.e)*- Here we can choose a nonsingular
point Q on E so that the following conditions are satisfied:
(i) E is defined by an equation u = 0 near Q.
(iiK.^eK.e)*.
Let v be a local parameter of E at Q. and choose an element v of &y.Q so that
v is the image of v in &e.q- Then (w. v) is a system of parameters of V at Q. and
the following relation holds:
dx Ady = ur+s~{((r£, + uS,u)r\v - (sn + unu)^v)du Adv.
where dc, = £,udu + £„dv and dn = nudu + nvdv. Therefore, we see that if we

11. MINIMAL ALGEBRAIC SURFACES
205
write Kv ~ f *Kw + J2; ai^i- ^en tne coefficient a, of £,. which appears among
Fy's. satisfies a,■ > r + s — 1 > 0. □
Set G = £7=i a,-£,-. Then (/ *ATn/ • G) = 0. Hence (AT,- - G • G) = 0. where
G ^ 0 because G > 0. As in the proof of the above lemma. (G • f *H) = 0 which
implies (G2) < 0. Thus. (Kv ■ G) = (G2) < 0. So (Kv • £,-) < 0 for some £,-. After a
change of indices, we may set £,- = £[. Since (£2) >0 by Lemma 11.3.1. £i is a ( — 1)
curve on V (Problem III. 10.2). By Theorem 11.1. there exist an algebraic surface V\
and a birational morphism <j\: V —» V\ such that a\ is the monoidal transformation
with center at a point Pi := a\{E\). In particular. o\\ V - E\ A V\ - {Pi} is an
isomorphism. So a birational mapping f\ : = f • <rf': V\ -+ W is a morphism
on V\ - {P\}. We shall show that f\ is a morphism at the point P\ as well. By
Corollary 11.2. ^V,.p, = PleeEi ^'O- ^mce &r.Q > @w.p- we have also ^V,.p, =
("We ^i'e - ^W- Namely. f\ is a morphism at the point Pi and f\{P\) — P.
Repeating the above argument finitely many times.we are reduced to the case where
dim/ _1 (P) < 0 for each P G IV. Thus. / can be written as a composite of a finite
number of monoidal transformations with centers at points. □
Corollary 11.4. Let f: V —> W be a birational morphism of algebraic surfaces.
Suppose dim/ ~[{P) > 0 for a point P of W. Then the following assertions hold:
(1)/ ~1{P) is a connected union of a finite number of nonsingular rational curves
E\ £„.
(2) £i E„ are linearly independent as elements »/NS(F)r.
(3) The intersection matrix ((£,■ • £7))i<,-y<„ is negative definite.
(4) Any two ofE\ E„ cross normally if they cross at all (Definition 8.4) and no
three of them have common points. (We simply say that E\ + ■ ■ ■ +E„ is a divisor with
simple normal crossings.) There is a ( — 1) curve among E\ £„. If one contracts
it. then there appears a (—1) curve among the images of the remaining curves under
the contraction. Contracting in this way ( — 1) curves, one by one. E\ +••• + £„ is
contracted to the nonsingular point P on W.
Proof. In view of Theorem 11.3. (1) and (4) are clear (compare Lemma 8.8
and Corollary 8.9). As explained in the proof of Lemma 11.3.1. (3) follows from
(2). We shall prove (2). Decompose / as a composite of monoidal transformations
/ = a„ ■ ■ -o\. where er,: F,-_i —» Vj. V = V0. and W = V„. Let Pt be the center of
a,-. and let £, be the proper transform of o/1 (P,) on V. Let Aj-\ (1 <;'<«) be an
ample divisor on V,-\. and let //,_i = (07_i • ■ • <j\)*(Aj-\). Then (//,-_ 1 • £,) > 0 and
(///-I • Ej) = 0 for each j < i. Suppose ai£i H 1- a„E„ = 0 for each a, £ R with
£1 £„ viewed as elements of NS(F)r. Taking the intersection of both sides of
this equality with Hn-\. we have an = 0. Considering successively the intersections
with //„_2 Hq. we get a„-\ = ■ ■ ■ = cx\ = 0. □
Let /: V —> W be a birational morphism. An (irreducible) algebraic curve on
V is called an exceptional curve of / if it is brought to a point of W by /. The
union of all exceptional curves of / is called the exceptional set. If a divisor D on
W has the irreducible decomposition D — £( a,C,. we define the proper transform
of D by / '(D) = £, a,-/ '(C,). where / '(C,-) is the proper transform of C, by /.
Let /: V —> W be a dominant morphism of algebraic surfaces. For an irreducible
divisor B on V. we define a divisor /tB on W7 by setting f „B = 0 if f(B) is a
point and /„/? = [fc(5): k(f(B))]f(B) if/(5) is an irreducible curve. For a divisor
^4 = J2i biB; on ^ m general, we define f*A = £. bjf*Bj. We call /*y4 the *#rm

206
III. ALGEBRAIC SURFACES
image of A by f. If f is a birational morphism. then A ~ A' implies f*A ~ f*A'.
(In fact. /*((A),-) = {h)w for A 6 k(V) = fc(»0.)
Lemma 11.5. Let f: V ^ W be a birational morphism of algebraic surfaces. Then
we have:
(1) For a divisor D on W, f *{D) — f'{D) is a divisor generated by exceptional
curves of f.
(2) For a divisor D on W and a divisor A on V. the projection formula (f *D -A) =
(D ■ f*A) holds.
(3) For divisors D.D' on W', if D ~ D'(D « D' or D = £>'), then f*D~
f *D'{f *D k, f *D' or f*D=f *D').
(4) f *: Pic(W) -> Pic(F) is injective. and
Pic(F) = /*Pic(^)ffi(£z[£,]j.
where {E\ E„} exhaust all irreducible curves belonging to the exceptional set of
f. and [E/] stands for the isomorphism class of$v(E,-).
(5) Similarly, we have
NS(*0R = /*Ns(»0RefX>[£;]).
Hence. p{V) = p{W) + n.
Proof. (1) is clear.
(2) We may assume that A is an irreducible divisor. If f(A) is a point then
(f *D • A) — 0. Hence, we may assume that B :— f{A) is an irreducible divisor on
W. Then A = f '(B). and by (1), f *B — A is generated by exceptional curves of f.
So [f *D-A) = {f*D-f *B). and we have only to show that (/*£>•/ *B) = (D ■ B).
Here we may apparently assume that D is irreducible. By definition,
(f*D.f*B)=X(0y)-X(M-f*D))-x(M-f*B))
+ x(<?v(-f*(D + B))).
and
(D ■ B) =x(&w) - X{&w{-D)) - x(@w(-B))
+ x{@w{D+B)).
Hence, it suffices to show that #(K/ *S?) = x{W.2?) for an invertible sheafs on
W. Meanwhile, for a spectral sequence
Ep2q = Hp{W.Rqft,f*2')^Hp+ci{V.f*2').
we have Rqf*f*3' = 21 <g> Rqf*@v (Problem 1.2.9). Since f is a composite of
monoidal transformations with centers at points, f*@v — @w and Rq f\&v = (0)
for each q > 0 by virtue of Lemma 10.3 (2) and the spectral sequence (1.2.29). Hence,
HP{W.SC) = HP{V.f*S?) for each p > 0. Thus, x{V.f*S?) =x(W,%).
(3) If D - D' = [g) with g e k{W) = k{V). then f*D - f*D' = (g).
Hence, f *D ~ f *D'. Suppose we are given a nonsingular algebraic curve C, and
an invertible sheaf S£ on W x C and points P. Q of C so that 5> = @w{D) and

11. MINIMAL ALGEBRAIC SURFACES
207
S'q = &W{D') (compare Definition 9.12). Put fc := f x \c: V x C -+ W x C
and S?(f) = f*c {ST). Then S?[f ] is an invertible sheaf onFxC such that SC{/) =
&v{f *D) and ^ ) = <fv{f *£>'). This observation entails that /*£>«/ *£>' if
D ~ D'. If Z) = £>' the projection formula in (2) above implies f *D = f *D'.
(4) As in the proof of (2). we have /»/ *SC = S? for an invertible sheaf on W.
Hence. /*: Fic(W) -+ Pic(F) is injective. It is clear that Pic(F) is generated by
f * Pic( W) and the isomorphism classes [£,] of ^,-(£,-)(1 < ;' < «). If f *D + a\E\ +
■ ■ ■ + a„E„ ~ 0 (for each a, <E Z). then (X)"_i Q'^')2 ~ ^. So «1 = ■ • ■ = a„ = 0 by
Corollary 11.4 (2).
(5) Suppose jV/ *Z) = 0 for a divisor D and W and an integer N ^= 0. Then
(JVD ■ 5) = {Nf*D ■ f*B) = 0 for any irreducible divisor B on W. Namely.
ND = 0. This implies that f *: NS( W)r —> NS(F)r is injective. The rest is proved
as in (4). □
Definition 11.6. Let V0 be an algebraic surface. Put 6 = {V; V is an algebraic
surface birationally equivalent to Vq}. For V. V £ &. define V > V if there exists a
birational morphism f: V^V. If V > V and V > V. then we have /j(F) = p{V)
by Lemma 11.5. Hence. V and F' are isomorphic to each other by Theorem 11.3.
Identifying V and V if they are isomorphic to each other, we can thereby define
an order in S by the relation V > V. In view of Lemma 11.5 and the fact that
p{V) < oo. we see that S satisfies the minimal condition. We call a minimal element
of S (not necessarily unique) a minimal model of V0. If Vo is a minimal model of
itself we call Vq a minimal algebraic surface.
What are the conditions for a given surface to be a minimal model and how
many minimal models exist? The following two results answer these questions.
Lemma 11.7. An algebraic surface V is a minimal algebraic surface if and only if
V has no (—1) curves.
Proof. Necessity. If there exists a ( — 1) curve E onV. then there is a contraction
a: V —> Vo by Theorem 11.1. Hence. V is not a minimal algebraic surface.
Sufficiency. Suppose V is not a minimal algebraic surface. Then there exists a
birational morphism f: V —> W which is not an isomorphism. By Theorem 11.3.
f is a composite f = an .. .o\ of monoidal transformations with centers at points.
Let E be the exceptional curve arising from o\\ V —> V\. Then E is a (-1) curve
on V. □
Theorem 11.8. Let V0 be an algebraic surface. If\NKy0\ ^ 0 for some positive
integer N. then Vo has a unique minimal model up to isomorphisms.
Proof. The proof consists of three steps.
(1) Suppose there exist mutually nonisomorphic minimal models V and V" of
Vo- Let p: V ■ ■ ■ —> V" be a birational mapping. By Corollary 8.11. there exist an
algebraic surface V and birational morphisms f: V —> V and g: V —> V" such
that g = p • f. where both f and g are composites of monoidal transformations
with centers at points. Let (£' be the exceptional set of f and let E\ E„ be its
irreducible components. Likewise, let <B" be the exceptional set of g and let Fi F„,

208
III. ALGEBRAIC SURFACES
be its irreducible components. By Lemma 11.3.1. we have
n
Ky ~ f *KV, + G. G = ^2 aiEi- "i > 0 for each '
m
Ky ~ g*Ky,> +H. H = ^2 h'Fi- b> > ° for each '■
Note here that \NK,>,\ ^ 0 and \NKy>,\ £ 0 by Lemma 10.7.
(2) We shall show that there exists a (-1) curve E with E C <£'. E ¢ <£". and
E n £" 7^ 0. Suppose £' = 0. Then V is isomorphic to V. and hence, does not
contain any (-1) curve. So g: V —> V" is an isomorphism as well. This contradicts
the hypothesis. Thus. £' ^ 0. Similarly. <£." ^= 0. Let £ be a (-1) curve contained in
(£'. If£c£". letffi: F -+ Fi be the contraction of E. Thengi :=g-CTf': F -+ F" is
amorphism (compare the proof of Theorem 11.3). Similarly. /( := / -(7,-1: Fi —> F'
is a morphism. Then replace F by Fi. Thus, we may assume that none of the (—1)
curves in <£' is contained in <£". Suppose E n S" = 0 for a (-1) curve E contained
in £'. Then F and F" are isomorphic to each other in the neighborhoods of E
and g{E). Hence. g(E) is a (-1) curve on V". and this is a contradiction to the
hypothesis that V" is a minimal model. Since £' ^= 0. £' contains a (-1) curve E.
which satisfies, by the above observations. E ¢ £". and E n £" ^ 0.
(3) Take £ as in (2). Let F := g(E). Then F is an irreducible algebraic curve
on V" with (F2) > 0. In fact, since E n £" ^ 0. g: F —> F" comprises a monoidal
transformation with center at a point of F. By Corollary 8.9. we know that (F2) >
(E2) = -1. The relation Ky ~ g*Kv„ + H given in (l) yields
-1 = {E ■ Ky) = (E ■ g*Ky„) + (E-H) = (F ■ Ky„) + (E • H).
Here we note that g.rE = F (Lemma 11.5). Since Suppi/ = (£". the condition on
E given in (2) above implies (E • H) > 0. Meanwhile, since \NKyn\ ^ 0 by (1). we
may write NKVi, ~ $3,-r;C,- (r,- > 0. C,: irreducible component). Noting here that
(F2) > 0. we conclude that {NKyn ■ F) = J2i n{Q ■ F) > 0. (In fact. (C,- • F) > 0 if
C,- ^ jf; (c,- • F) = (F2) > 0 if C,- = F.) So we have (F • AT,'») + (F • H) > 0 which
is a contradiction. This contradiction is caused by our assumption that Vq has two
nonisomorphic minimal models. The theorem is therefore proved. □
Definition 11.9. If an algebraic surface F has a unique minimal model Vq up to
isomorphisms, then we call F0 the absolutely minimal model or simply minimal model
of F. When we call the absolutely minimal model Fo the minimal model of V. we
call a minimal model in Definition 11.6 a relatively minimal model to distinguish one
from the other.
Definition 11.10. Let F be an algebraic surface. A quantity associated with F
is called a birational invariant if it does not vary for various birational models of F.
As one of such invariants, we define the Kodaira dimension n(V) as follows. For a
canonical divisor Kv of F. set N(F) = {« <E N: \nKv\ ^ 0}. We define n(V) = -oo
if N(F) = 0. If N(F) f 0. we define k{V) by
k{V) = max{dim<D|B/i:r.|(F): n £ N(F)}.
Accordingly. k(V) takes one of the values —oo. 0.1.2.
Theorem 11.11. k(V) is a birational invariant.

11. MINIMAL ALGEBRAIC SURFACES
20<)
Proof. In view of Corollary 8.11 and Theorem 11.3. we have only to show that
k{V) = k{Vo) if a: V —> Vo is the monoidal transformation with center at a point
P. Let E := a~\P). Then Kv ~ a*Ky„ ~ E (Lemma 10.3). If \nKVl)\ f 0.
then \na*KVil\ + nE C \nKv\. So |«A"k| 7^ 0. Conversely, if \nKv\ ^ 0. then
\nKy0\ ^ 0 because atKy ~ £>„. We therefore have N(F) = N(F0). Furthermore.
H°(V.&(nKv)) = H°(Vo.0{nKVo)). This entails k(F) = k(F0). D
Lemma 11.8 is equivalent to saying that V has the absolutely minimal model
provided k(V) > 0.
III.ll. Problems
Let P\ P,.(1 < r < 8) be points of P2. and let a: V —> P2 be a composite
of monoidal transformations with centers at Pi P,.
1. Let C be an irreducible algebraic curve of degree d on P2. let w, = ju(C;Pj).
and let C := o'{C). Show that the following equalities hold:
(C2) = d2-J2mf- (CKy) = -3d + J2mi-
/=1 /-1
2. With the notations of the problem 1. C is a (-1) curve if and only if
r r
d1 — 2~]mf ~ ~3d + z2m' = ~^-
/=1 /=1
With the additional conditions that m, > mi > ■ ■ ■ > w,- and
(rf 22)-^^^ +0^1-
show that (d:m\ mr) is one of the following 6 collections:
(1; 12). (2:15). (3; 2.16). (4;23. 15). (5.26.12). (6: 3. 27).
where (1; 12) stands for (1:1.1.0 0) and the other expressions are similar
/•-2
abbreviations.
(Hint: We have only to consider the case r = 8 by putting mr+\ = • • • = m% — 0.
Then derive the relation 18 = X^=i(w/ _ I)2 + XL</(W< ~ mj)1)
3. Assume that any one of the collections [d;m\ mr) in problem 2 above is
realized uniquely by an irreducible algebraic curve passing through the points
Pi P,. We. therefore, assume that for any substitution (P,, P,- ) of
(Pi P,), there is a unique irreducible algebraic curve of degree d with the
assigned multiplicity m;- at P,- (for each j. 1 < j < r). Then show that the
number of (-1) curves on V is given by the following table.
Table
r
N
1
1
2
3
3
6
4
10
5
16
6
27
7
56
8
240

210
III. ALGEBRAIC SURFACES
We say that the points P\ P, are in general position if Pi P, satisfy
the condition required at the beginning of this problem. An algebraic surface
Vr obtained by a composite of monoidal transformations with centers at such r
points on P2 is called a del Pezzo surface together with P2 and P1 x P1.

CHAPTER 12
Ruled Surfaces and Rational Surfaces
Let B be a nonsingular algebraic curve and let Jbea locally free ^-Module
of rank r. Choose an open covering ii = {[/;J;.eA of B such that W\yi is a free
&v>-Module for every X e A. that is. F\u, = efu,e[A) H + &\j-e\':) for a free basis
{e[^ e^}. For X.fi £ A, on U^ : = [/;. H [/„. we can write
(a) e^=J2«ijef- -0^(¾.¾).
./ = 1
where ^. := (a,-y) is an (r x r) matrix with entries in T{U^.c?b) and there is a
relation Ar;_ = Ar/l ■ A^; on [/; ,, := [/;_ fl^n [/,. for l./x. v S A. Moreover. Ay,_ is
the identity matrix of size r. We call {A^;.} the transition matrices of F~ with respect
toil.
We define a symmetric &B-Algebra £'(3r) generated by & as <S'{&~) =
®,>o<^'(-^)- where £'(&~) is the /th symmetric tensor product of ^". With the
above notation, we have r{U;_.S'(F)) = r([/;..<fB)[e{/') ^1( = a polynomial
ring in r variables over T{U;..$b))- If we assume [/;. to be an affme open set.
Proj(r(Ux,£'{&■))) is identified with Uk x F~' and {e({-] e,w} is its system of
homogeneous coordinates. For each L/i £ A. [/; x P'_1 and [/^ x Pf_1 are glued
up by the relation (a) above and a 5-scheme Proj(oS"(^)) is. thus, obtained. We
denote it simply by F(F) and call it the projective bundle associated with fF.
Since {e[A' e, '} is a system of homogeneous coordinates of U; x P''_1. we
define an open set [/;.., of [/;_ xpr~l consisting of points for which e\'"' ^ 0. We then
have [/;. x P'"1 = \Jj=x [/;..,-. Furthermore. {e\ @u,,}\<i<r are patched together to
form an invertible sheaf $;.(l) on [/;. x Pr_1, and the {<^;.(l)};.eA are patched together
to form an invertible sheaf ^.^(1) on P(^). We call <^p(^)(l) the tautological
invertible sheaf on P(.!F). The transition function of ^^(l) on [/; ,■ n [/^.y is given
by the relation (a) as ^=1 a^e^/ef • Let /: Pf^F) —> 5 be the structure morphism
as a 5-scheme. Then there is an isomorphism ip: & —> /*(^p(^-)(l) which is given
by (<pk)(e,U)) = e,U) (l<i<r.AeA).
Lemma 12.1. Le? B be a nonsingular algebraic curve, and let SF be a locally free
&b-Module of rank r. Then the following assertions hold:
(1) There is a sequence of locally free &B-Modules
(0) = 5¾ C 2~\ C • • • C Wt = S?
such that SFi/OFi-x is an invertible sheaf on B for 1 < i < r. If we set det^" =
211

212
111. ALGEBRAIC SURFACES
0/=1 &~i/&~i-\- det^" is determined independently of the choice of the above sequence
of locally free &B-Modules.
(2) If B is a projective algebraic curve, then we have
x{B.^) = deg(det^) +rX{tfB)-
Proof. (1) Let K = k{B). The generic fiber P(^),, := P(^) xB SpecK of
the structure morphism f: P(Sr) —> B is isomorphic to P^_1. so Y'Hf)n has a K-
rational point. Let £ be a A'-rational point, and let S = {£} be the closure of c, in
P(^). Then f \s: S —» B is a birational proper morphism. Since B is a nonsingular
algebraic curve, f \s is an isomorphism. The inverse (f |s)~' gives rise to a closed
immersion a: B ——> S <-+ P(^) such that f • a = idB. A morphism a: B —> P(.!F)
with /' • <T = ids or its image S =a{B) is called a section (or a craw section) of /. If
we identify 5 with _6. ^p(^-)(l)|s is an invertible sheaf on B. With the defining ideal
sheaf S of the closed subscheme S of P(^). we have the following exact sequence
O - S ® ^,(1) - <W1) ^ *p(5r)(l)U - 0.
Thence, we obtain an exact sequence
o - fM ® ^)(1)) - ^ -^ ^P(^)(i)U'.
where (// = /*(rc). We shall show that ^ is a surjection. Replacing, if necessary,
the open covering il by a finer covering using the notation prior to Lemma 12.1. we
may assume that for every X £ A. o'U-,) c U-,..-, (for some i). Suppose a(UA) C
U,A for some X. Then {<fn^{\)\s)u, = (9i;{e^\s).e\% = ai(e^\s) with a,- e
r(t/;..(^) (1 < ' < >")• and ^Ij,- maps e^.ep to e^l^.a, • (e,W|s). respectively.
Hence. (//|^ is surjective. Since A is arbitrary, y/ is surjective.
Set &r := 5T ^._i := f^S ®@?m(1)). and 2*,. := /fp^l)^-. We then have
an exact sequence
0 —> ^._i -+^--^,- ->0.
Since 5^ and .2,- are locally free <fg-Modules of rank r and 1. respectively. ,9>_i is
a locally free <f#-Module of rank r — 1. Repeating the above argument, we obtain
exact sequences of locally free (¾-Modules.
(b) 0-^--1-^--^.2/-^0 (l</<r).
where i?o = (0) and every .2,- is an invertible sheaf.
If we replace the open covering H by a finer covering if necessary, then we may
choose a free basis {e\A' e,-} of & on U;. in such a way that {e\A' e) }
is a free basis of fF\ j u, ■ Then the transition function A^k of JF on t/^ given by
the relation (a) is a lower triangular matrix, and its diagonal components are the
transition functions of 5C\ .2,-. Hence. det(^;J is the transition function of the
invertible sheaf S?\ ® ■ ■ ■ ® S?r. So det & : = SC\ ® ■ ■ ■ ® .2,- is independent of the
choice of the sequence of Modules (0) = 3¾ C &\ C - - • C &r.
(2) The exact sequence (b) yields xffi) = x(&~i-\) +/(-2/)- So we have
r i-
X&) = 5Zx(^) = XXde^+?M} = deg(det^) + rX{SB). U
/=i /-i
Lemma 12.2. Let & be a locally free Module of rank r over B = P1. Then there
exist integers d\ dr such that & = &{d\) © • • • ®@'dr).

12. RULED AND RATIONAL SURF-ACES
213
Proof. We note that Pic(P') = Z[^(l)] by virtue of Example II.5.17.1. Though
we give a proof below only in the case r = 2. it is not difficult to extend the proof
to the general case. The proof consists of three steps.
(1) By Lemma 12.1. there exists an exact sequence
(c) 0 -> J5?i -* J- A Se2 -* 0. S?, = 0{dj). i = 1.2.
If EF contains an invertible sheaf SC as an <9B-sub-Module, then degJ? <
max{d\.d2). In fact, if ^{5?) ^ (0). y/\s'- SP —*■ -¾ corresponds to a nonzero
element of H°{B.S?2 ® 2f~x). So deg(^2 ® 2f~x) > 0. Namely. d2 > deg5f. If
y/(S?) = (0). then^C ^. Hence. H°(B.S?\ ®2"1) ^ (0). So </i > deg^f. Thus.
degSf < ma\(di.d2).
(2) Choose an exact sequence (c) so that d\ — deg J?i is the largest among all
exact sequences of the same kind. The largest value exists because of step (1). We
shall prove by reductio ad absurdum that d2 < d\. Suppose d2 > d\ on the contrary.
Replacing &~ by EF ® &{—d\ - 1). we may assume that d\ = —\ and d2 > 0. The
exact sequence (c) then gives rise to a cohomology exact sequence
0 - H°(B.r) -^H H0(B.Sr2) - H\B.<9{-\)).
where H°(y/) is an isomorphism because H](B.<&{ — 1)) = (0). Since d2 > 0.
Ha(B.S?2)j= (0). So Ha(B.&) j= (0). Let a be a nonzero element of H\B.F).
With the notation before Lemma 12.1, write ct|[/; = s\e{ + s2e\ with ^.¾ £
T{U)_,ffB). We may assume si 7^ 0. Since P^^t/, = Ui x P1. define a rational
mapping t : B -»P(^) as the graph of a mapping ge^i-* (.^(2)- ~s'i (2)) £ pl- If
77 is the generic point of B. the closure {t(^)} gives rise to a section of/: P(5r) —> 5.
Namely, t is extended to a morphism t: B —> {z(rj)} ^-> P(^). This morphism t
entails the decomposition Sf in the form of an exact sequence
where ^\ and ^2 are invertible sheaves, and where ^\ = f*{S ® @p(&){\)) with the
defining ideal sheaf S of the section S : = r{B). In particular, a\U; € r(C/;..^i).
Since Ae A is arbitrary, we have <r e H°(B.&]). So deg^i > 0. This contradicts
the choice of the sequence (c) for which d\ =-1 is the largest.
(3) The exact sequence (c) entails an exact sequence
0 -► &\ ® 2?2X -> & ® S?2-y ^Z2® S?~x -* 0.
where y/' : = y/ <g> j?9~'. Consider the cohomology exact sequence
0 -* H°(B. SCi ® S?^)-> H°(B, & ®2,1~x) ^X H°{B. S?2 ® 2>^)
^h\b.%\®%-x).
where Hl{B.S?\ ® Z2y) = H\B.@{d{ - d2)) =* H°{B.&(-2 - </, + d2)) = (0).
(In fact, since d\ > d2, —2 — d\ + d2 < 0.) H°(y/') is, therefore, surjective. Since
HQ(B.Z2 ® .¾-1) = Hom(^2.^2) there is an element a € H°(B.S?2 ® 2f~x)
corresponding to the identity mapping id: S?2 -+ 5f2. Take a € H°{B.!F <8> -¾-1)

214
III. ALGEBRAIC SURFACES
so that a — H°(i//'){a). Let /?: .2¾ -+ SF be an ^-homomorphism corresponding
to a. Then y/ •/? = id. So & decomposes as & = S\ 0/7(.2¾). D
The following result is due to M. Noether.
Theorem 12.3 (M. Noether). Let p: V -+ B be afibration on an algebraic surface
V. Suppose general fibers of p are nonsingular rational curves. Then V is birational
to B xP1.
Proof. The proof consists of three steps.
(1) Let Vn (resp.. Vh) be the generic fiber (resp.. a general fiber) of p. Vn is an
algebraic curve defined over K = k(B). and Vf, is isomorphic to P1 by hypothesis.
We shall show that xi&r,,) ■= H)-o(-l)''h'(v>i■&!'„) = x(&vh). In the subsequent
argument, we use the condition that Vh is an irreducible algebraic curve, not the
condition that Vh = P1. Let g: W -+ T be the morphism obtained from p as the
base change by a flat morphism T := Spec ^./, -+ B. We have R]g*@w — (Rlp*<fv)h-
Writing mBh = t@Bh- we have an exact sequence of ^-Modules
0 -► t&w -► @w -+ &Vh -+ 0.
Hence, we obtain an exact sequence of <9 : = $B./,-Modules:
0 -+ gj&w -+ g*@w -+ H°(Vh.ffVh) -+ R]gJ0w
-+ Rxg*@w -+H\Vh.@Vh)-+ R2g*t@w - R2g^w - 0.
Since g is a proper morphism. R'g*@w is a finitely generated if-module (II.4.18)
and R'g*{tffw) = tR''g*@w. Hence, R?g*@w = (0) by Nakayama's lemma. Since
g*@w — & '.= @Bb- the above cohomology exact sequence yields the following exact
sequence:
0 - tR'g^w -+ Ryg^w -+ H\vh.@Vl) -+ 0.
Hence. Hx{yh.@y,) = Rxg*@w ®s k — Rxp*0v ®@B k{b). where the second
isomorphism follows from (II.4.7.1) because g is the base change of p by a flat morphism
T -+ B. Similarly, noting that SpecK -+ B is a flat morphism. we obtain an
isomorphism
{R}p^v)n = R1(pk)^v„ = H\vn.@Vi).
where pK = p ®0B K. Let J~ be the torsion sub-Module of Rxp*@v. (For an
affine open set U. let !TV be the &v -Module obtained from the torsion submodule
of T(U.R]p*@v)- Then the {5¾} are glued together to form a coherent ^-Module
!T.) Note that Supp^ is a finite set. Choose a point b so that b ¢ Supp^. Then
rank(^V*^)/> = rankCR'/^i'V Hence. hx{Vh.@Vh) = hx(yn.&Vl). This implies
x(&v„) = x(f_K,)-
(2) Let K be an algebraic closure of K. and let V n = Vn <giK K. Since K(Vn) =
k{ V) is a regular extension of K (Lemma 9.1), V n is an irreducible reduced algebraic
curve defined over K. Noting that H\Tn.@y) = Hx{Vn.@Yi) <giKK = (0), we
have pa{Vn) = 0 (Lemma 10.5). By Lemma 10.4, Vn is a nonsingular algebraic
curve of genus 0. Hence. Vn = P~ (Theorem 7.6). If we set Q. = O^/jc. we have
Q. <g>K K = £ly ,-g. These observations give
A0(K,.n-1) = A0(K),.(IVv/X)-I)=A0(P1.(np1)-I)=A0(P1.^(2)) = 3.
Let {/<,■ /1./2} be a ^-basis of H°(Vn. ft"1), and let ^: P, . P2K be the rational

12. RULED AND RATIONAL SURFACES
215
mapping associated with {/0/1/2)- Then ip is denned over K and_v? ®k K =
d>l_K ,|. where KPi is the canonical divisor of V\ = P^. Since <p ®K K is a closed
immersion. ^ is a closed immersion as well. (Prove this as an exercise.) Since
<p*@P2(l) = Q~l and deg(nr,/^)_1 = 2. f(V,,) is a conic (= a curve of degree 2)
in P^. Namely, with homogeneous coordinates (X0.X[.X2) of P|. ^(^) is defined
by a homogeneous equation Q{Xq. X\.X2) = 0 of degree 2 with coefficients in K.
(3) Here we refer to a theorem of Tsen. But we need some preparation before
stating the result.
Let A' be a field. Take an arbitrary homogeneous polynomial f{X\ X„) of
degree d with coefficients \vlK. If the equation f(X\ X„) = 0 has necessarily a
root («i a„) (for all a, £ K) other than (0 0) whenever n > d. we call K
a Q-field. □
Lemma 12.3.1 (Theorem of Tsen). Let k(B) be the function field of an algebraic
curve B defined over an algebraically closed field k. Then k(B) is a C\-field.
With the previous notations K = k{B). K is a C\-field. Since 3 > 2. the quadratic
homogeneous equation Q(X0. X\. X2) = 0 has a root (ao.a1.a2) other than (0.0.0)
with all a, £ K. Namely. <^(K/;) has a /^-rational point. Let C = ip(Vn), and let
P be a ^-rational point of C. Then h°{C.0c{P)) = h(){C.@T{P)) = 2. where
C ®K K = P^. The morphism ¢1/)1: C -> P^ is an isomorphism and is defined
by means of a A'-basis of H°(C.&C(P)). So (D^ is considered as the base change
ip ®k K of an isomorphism of iC-schemes <p: C —> YXK. Namely, Vn is isomorphic
to Pjj- over K. Thus. V is birational to P'xfi, □
Definition 12.4. An algebraic surface V is called a ruled surface if F is birational
to B x P1, where 5 is a nonsingular complete algebraic curve. A fibration p: V —> S
is called a P1 -fibration if general fibers are isomorphic to P1. The reducible fibers and
nonreduced fibers of/? are inclusively called singular fibers. We call p a P1 -bundle or
Pl-fiber bundle if every closed fiber of/j is isomorphic to P1. Note that the generic
fiber Vn of a P1-fibration p: V —> B is isomorphic to P1 over k(B).
Lemma 12.5. Let p: V —> B be a P1 -fibration. Then p has a section. Hence, there
are no multiple fibers in p. Let F = Y?i=\ ai^> (ai > 0- Q: irreducible component) be
a singular fiber of p. Then the following assertions hold:
(1) There exists a (—1) curve appearing as an irreducible component ofF. Suppose
C\ is such a (—1) curve. If a] - 1, then there exists another (-1) curve among the
irreducible components of F.
(2) Let o\\ V —> V\ be the contraction of C\. Then the rational mapping p\ =
p • o-f : V ■ ■ ■ —* B is a morphism and is a P1 -fibration for which F\ := (<7\)tF is a
fiber. If F\ is a singular fiber of p\. we obtain also a P1-fibration by contracting a (-1)
curve which is an irreducible component of F\. Repeating the contractions of this kind,
the direct image of F becomes finally a fiber of a P1-fibration which is isomorphic to
P1.
(3) Every C, (1 < £ < n) is isomorphic to P1. Furthermore, no three components of
F have a common point, and there appear no loops among the irreducible components
of F. (Suppose, for example, irreducible components C\ C, satisfy the condition
[Ct ■ Ct) = \for {i.j) = (1.2). (2.3) (r - 1. r). (r. 1) and (C, • cj) = 0. otherwise.
We then say that C\ Cr form a loop.)

216
III. ALGEBRAIC SURFACES
Proof. (1) The generic fiber Vn of p is isomorphic to P1 over k(B). Hence. Vn
has a £(5)-rational point. It follows from Lemma 12.14 that p has a section and p
has no multiple fibers. Let F be a closed fiber of p. Suppose F is irreducible and
reduced. Since F is algebraically equivalent to a fiber, say G. which is isomorphic
to P1. we have (F • Kv) = (G • Kv) = -2- (G2) = -2. Since (F2) = 0. we have
pa{F) = 0. By Lemma 10.4. F is then isomorphic to P1. Suppose F is a singular
fiber. Write F = £"=, a,-F,-. Then r > 2. By Lemma 9.14. (C2) < 0 for each i.
Since (F ■ Kv) = -2. (C, • Kv) < 0 for some i. say ? = 1. Then (C,2) < 0 and
(Q ■ ATr) < 0. So C[ is a (-1) curve (Problem III. 10.2). Suppose ax = 1. Then we
have
-2 = (F • Kv) = -I +J2a'(C' • Kv)-
So there exists j > 1 such that (C,- • AV) < 0. This C; is then a (-1) curve. Thus,
the assertion (1) holds.
(2)Let/'i:=ai(C1). By Corollary 11.2. ^,v/>, = flgec, ^'2- Since <?v.e > 0B.h
with Z> = /7(F). we have @VvPx > ^flft. This implies that p\ is regular at the point P\.
If F\ is reducible, we can repeat the contractions of this kind. Finally, we obtain an
irreducible reduced fiber in a P'-fibration. This fiber must be isomorphic to P1 as
we have shown above.
(3) Conversely, a singular fiber F of p is obtained by repeating the monoidal
transformations with centers at points (and infinitely near points as well) on a fiber
of a P'-fibration which is isomorphic to P1. Then every irreducible component of
F is isomorphic to P1. and no three of the irreducible components have a common
point. It is clear by this construction that there appear no loops among irreducible
components of F. □
Let V be an algebraic surface. We call V a rational algebraic surface (or a rational
surface, for short) if the function field k{ V) is a purely transcendental extension
of k. Suppose V is a ruled surface birational to B x P1. Then k(V) is a purely
transcendental extension of k(B). By virtue of Luroth's theorem (1.1.36). V is a
rational surface if and only if B is a rational curve. If B is irrational, we call V an
irrational ruled surface. We note that a rational surface is a ruled surface.
Lemma 12.6. Let V be an irrational ruled surface. Then V has a P]-fibration. A
minimal model of V has a P1 -bundle structure.
Proof. There exists a birational mapping 9: V ■ ■ ■ —f B x P1. where B is an
irrational algebraic curve. By Corollary 8.11. there exists a birational morphism
a: W -* V such that x := 0 • a: W —> 5 x P1 is also a birational morphism.
Let p: B x Pl —> B be the natural projection. Then q := p • x: fF-^-Bisa P1-
fibration. In fact, since x is a composite of monoidal transformations with centers at
points (Theorem 11.3). almost all fibers of p remains intact under t. As the birational
morphism a: W —> V is also a composite of monoidal transformations with centers
at points, let E be a (-1) curve on W which is contracted to a point by a. By Luroth's
theorem, the morphism q\E: E —► B is not dominant. Namely, E is an irreducible
component of a closed fiber of the P'-fibration q\ W —> B. Let o\: W —> W\ be the
contraction of E. Then q\ := q • a~l: W\ —> B is a P'-fibration by Lemma 12.5. The
mapping a ■ cj"1: W\ —* V is a birational morphism. If it is not an isomorphism,
we repeat the foregoing arguments. Hence. V has a P'-fibration p: V —► B. If/?
has reducible fibers, we contract successively (-1) curves contained in the closed

12. RULED AND RATIONAL SURFACES
217
fibers as irreducible components to obtain an algebraic surface with a P1-fibration
po '■ I7o —> B such that all closed fibers are irreducible fibers. Then p0 is a P1-bundle.
Furthermore, there are no (-1) curves on Vo. So by Lemma 11.7. V0 is a (relatively)
minimal model of V. □
Lemma 12.7. Let p: V —> B be a P1 -bundle. Then there exists a locally free
Module & of rank 2 defined over B such that V is isomorphic to P(^") as B-schemes.
Proof. Let q be a £(.#)-rational point of Vn, and let S = {£}. Then S is a section
of p. We shall show that & : = p*@y{S) is a locally free ffg-Module of rank 2. For
this purpose, let b be an arbitrary closed point of B. and let t be a local parameter
ofBatb. Since F/, = P1. we have ffy{S) ®&Vh = ^Pi(l) which we denote by ^,,,(1).
Let g: W —> 7" be the base change of p by a flat morphism T : = Spec ^./, -+ /?.
We have an exact sequence
o-kMs)-<Ms)^^ (1)-0.
where ^V(S) = ^(S1) ®^r $V. Set ^ : = ^./,. Then this exact sequence entails an
exact sequence of finitely generated & -modules
0 - tg^w{S) - g.0w{S) -+ H°(Vh.ffVh{l)) - tR}g.0w(S)
->Rxg^w(S)^H\vh.0Vlt{\))-
Since //1 (F,. &Vh{\)) = (0). we have fl'g*^S) = (0) by Nakayama's lemma. Thus,
we obtain an exact sequence
0 -+ tg.0w(S) -+ g*^(S) - //0(^.^,,(1)) - 0.
Since g*0w{S) = p*&v{S) ®av ®w- we have after all p*@v(S) ®ffB k{b) =
H°(Vh.@Vh(l)). Since b is an arbitrary closed point of B. we know that p*&r(S) is
a locally free ^-Module of rank 2. On the other hand. p*&v{S)n = H°(Vr@Vii{l)).
Let {ei. e2} be a fc(5)-basis of S^. and let 0n: Vn — P(5r), = Proj fc(5)[ei. e2] be the
associated k(B)-rational mapping which is in fact an isomorphism. Regard 9n as a
birational mapping 0: V > P(F") of 5-schemes. Namely, if f: P(^) -+ B is the
structure morphism of the 5-scheme P(^). then p = f -0. Let 6 be a closed point of
5. With the above notation, we see that we can choose a free basis [d\. d2} of a free
ff-module g*@w{S) of rank 2 so that {0-1.0-2} is a fc-basis of //°(^/,.^(1)), where
o, = o,- (mod tg^w(S)). i = 1.2. The restriction of {oV oS} onto F, gives rise to a
A:(5)-basis of H°( Vn.^,/,(1)). Let 9:W -> P(^,) := Proj ^[oVo2] be the morphism
associated with {01. o2}. Then 0^ coincides with 6n up to a projective transformation
of P|.(£). Moreover. 0/, is the morphism F, -+ P(^ <8>k(b)) associated with a fc-basis
{01.02}. Hence, we know that 9 xBT = 9. Since b is arbitrary. 9 is an isomorphism.
We can define 0 as follows. Let ii = { f/;.};.eA be an open covering of Z? such that
f?\v, is a free &v-Module for each X e A. Let {e\'] .e^} be an ^,-free basis of Sr\U/.
Then we can define a rational mapping 9;: p^l(U;) —> Y{^\y) = Proj^/.[e^'. ej ]
over f/;. naturally by 2 ^ (<?i (2)-4 (2))- The rational mappings 5; and 9M are
transformed to each other by the projective transformation induced by the transition
matrix between {e,. ef'} and {e, . e, }. Thence we know that the {0;.};.£a are
glued together to form a rational mapping 0: V ■ ■ ■ -+ P(^") over 5. The previous
argument entails, apparently, that 9; is an isomorphism for each 1 e A. O

218
III. ALGEBRAIC SURFACES
Lemma 12.8. Let &~ be a locally free @B-Module of rank 2. let p: Y*{&~) —> B be
the P1-bundle over B associated with 3~. and let @v^){\) be the tautological invertible
sheaf. Then we have Pic(P(5r)) = p* Pic(B) ® Z[^P(y)(l)].
Proof. Let n be the generic point of B. and let .K =k(B). Write 5^, =Ke\ +Ke2.
Let ^ be a /^-rational point of P(^")f/ = Pj^ which is given as (0.1) in terms of
homogeneous coordinates (e\. e2). and let S = {£}. Then S is a section of p. Now
let D be an irreducible divisor on P(^"). If p{D) — B. then Dn : = D xB SpecK is an
effective divisor on Pj^ which is defined by a homogeneous equation G{ei.e{) = 0
with coefficients in K, Let d — {D • F) for a general fiber F of p. Then d = deg G.
Let g : = G{e\. e2)/<?f • Then g is viewed as a rational function on P(^) defined over
k. Moreover, the divisor D - dS - (g) has no irreducible components which meet
Y{9')n. Namely, we can write
/■
D-dS- [g) =S^aip-\bi). a, € Z. b,- G B.
/=1
This implies that Pic(P(^")) = p* Pic(5) ® Z[&P{sr)(S)]. Meanwhile. ^P(y)(l) and
&V^W){S) as restricted to P(^), are isomorphic. Hence. <fP(5r)(l) = ^^ (S) ®p*£?
with Jz? G Pic(-B). The assertion follows from this description. □
Corollary 12.9. Let ^.% be a locally free (9B-Modules of rank 2. Then P(5r)
is isomorphic to P(^) as B-schemes if and only if§ = 5*" ® 3f with SC £ Pic(S).
Proof. Necessity Let f: P(^F) —► £ and g: P(^) -^ B be respective
structure morphisms and let ip: P(^) -+ P(^) be a S-isomorphism. By Lemma 12.8.
¥>*<V)(l)s<?P(^)(l)®/*^for^€Pic(*). Hence./^^(^(1)=/-,(^(^(1)¾
f*S?) = SF ® 5?. Meanwhile, if we set y/ = <^_1. the pullback homomorphism
if*: Pic(P(^)) —> Pic(P(^)) coincides with the direct image homomorphism
^: Pic(P(S?)) - Pic(P(^)). Hence. f,<p* = />, = g,. So /^^)(1) =
g*^P(s-)(l) = ^. Thus. ^ = & ® J?.
Sufficiency. This is clear from the construction of P(^") and P(^). □
Define a P1-bundle F„ over P1 as F„ = P(^f © <f («)). where « is a nonnegative
integer, & = @v\ and <f(w) = @v\{n). We call F„ the Hirzebruch surface of degree
n. Put SF = @ ®@(ji) and choose an open covering 11 = {t/;i};6A of P1 such that
&{n)\v, ^ ^e{A). a free ^-Module for each AgA. Then 5^ = ^,e0 + &V;ef
and (e0-e, ) are homogeneous coordinates of P(3r)\u/ = Proj^^o-e]\ ]■ where e0
is taken independently of X e A. Let M;t (resp.. M'k) be the section on Uk defined by
q = 0 (resp., e0 = 0) relative to homogeneous coordinates (<?o- e, ). Then {M;};l6A
(resp.. {M]};eA) are patched together to give a section M (resp.. M'). (We say that
M (resp., M') is the section defined by the natural projection !F \— @ @@{n) to
& (resp.. @{n)) If 7i: F„ —> P1 is the structure morphism of F„. M (resp.. M') is
defined on ti"1 (Uk) : = U, x P1 by the equation e[;)/e0 = 0 (resp.. e0. e\X) = 0). Let S
be the defining ideal sheaf of M in F„. Then (J^/J72)^, = @v, • {e[x)/e0). Hence,
if we identify M with P1 via %. JI J2 = ^P,(«). Since (M2) = deg(^/Jr2)v.
we have (M2) = -«. Similarly, if / is the defining ideal sheaf of M'. we have
f /f1 ^ (fp,(-n) and (M'2) = «. Furthermore. (M ■ M') = 0 by the definition of
M and M'. Extending the above observation, we obtain the following result.

12. RULED AND RATIONAL SURFACES
219
Theorem 12.10. (1) F„ {n = 0.1.2 ) exhaust all ¥{-bundles over P1 up to
isomorphism.
(2) Let n: Fn —> P1 be the Hirzebruch surface of degree n. let M.M' be the
above-defined sections of n. and let I be a general fiber of n. Then we have:
M'~M+nl. (M-M') = 0. (M2) = -n. (M-/) = 1,
Pic(F„) = Z[/] © Z[M]. KF„ ~ -2M - {n + 2)/.
In the case n ^ 0. an irreducible divisor C on Fn with (C2) < 0 is identical with the
section M. {We call M the minimal section of Fn.) In the case n = 0. Fq = P1 x P1.
Proof. (1) Let V be a P1-bundle over P1. Then V = P(^) for a locally free
^-Module & of rank 2 (Lemma 12.7). By Lemma 12.2. & = &(dx) © &(d2)
for integers </i,</2. By Corollary 12.9, V = P(<? © ^(«)). where n = d2 - dx if
d2 >d\ {n — d\- d2 if <i] > <i2)- Namely, every P'-bundle over P1 is isomorphic to
a Hirzebruch surface over P1.
(2) By Lemma 12.8, M' ~M + rl. Since (M2) = -n. (M • /) = 1 and (M'2) = «'.
we have n = —« + 2r. So r = n. Let C be an irreducible divisor on F„. Write
C ~ aM + /;/. Here a = (C • /). and / moves in a linear pencil |/|. Since C is
irreducible, we have a > 0. If a = 0. then C e |/|. Suppose (C2) < 0. Then a > 0.
Since (C2) = a(2ft - an) < 0. we have 2ft < an. Then (C • M) = b - an < 0. So
C = M. The other assertions are readily ascertained. □
We shall state the next result without proof.
Theorem 12.11. Let V be an algebraic surface.
(1) V has Kodaira dimension k(V) = —oo if and only if V is a ruled surface.
(2) Relatively minimal models of an irrational ruled surface are P1 -bundles.
(3) Relatively minimal models of a rational surface are exhausted by P2 and Fn (n =
0.2, 3,...). Hence, they are P1 -bundles except for P2.
We proved the assertion (2) in Lemma 12.6. Let p: V —> B be a P1-bundle,
let / be a closed fiber, and let P be a point on /. Let a: V —> F be the monoidal
transformation with center at P. let £" = a~l(P), and let /' = a'(I). Then /' is a
(—1) curve on V' which is therefore contractible. Let t: V —> V be the contraction
of /'. Then p := p • a • t~' : V —> S is again a P'-bundle. We denote by elm/- the
birational mapping t • a'1: V ■ ■ ■ —> V and call it the elementary transformation with
center at P. A P1-bundle /¾: Fb —+ 2? is called a trivial P'-bundle if it is isomorphic
to p\: B x P1 -4 B (/7i being the natural projection) as 5-schemes.
The following theorem will explain the significance of elementary
transformations.
Theorem 12.12. Let B be a nonsingular complete algebraic curve. Then every P1-
bundle over B is obtained, up to isomorphisms, from the trivial P1 -bundle V0 = B x P1
by applying a finite number of elementary transformations.
Proof. We consider only the case B = P1. Regard 7¾ = P1 x P1 as the trivial
P'-bundle over P' be the first projection (x.y) i—> x. Let / be a closed fiber, let P
be a point on /. and let M be the fiber of the second projection 7¾ = P' x P' —*
P1, (x.y) i—> y which passes through the point P. With the above notations a and t,
M := t(o'(M)) is a (—1) curve on the P'-bundle F0 and a cross section as well. By
Theorem 12.10, Fq = F\ because M is the minimal section of Fq. Next take a point

220
III. ALGEBRAIC SURFACES
P on the minimal section of F\. Then elnip.Fi = Fi. Similarly, if n > 0 and M„ is
the minimal section of F„, we have elmp F„ = Fn+\ (if P € M„) and elm/. F„ = F„_i
(if/^M„).
The minimal section M\ of Fi is a (-1) curve, and the contraction of Mi produces
P2 out of F\. a
111.12. Problems
1. Let B be a nonsingular complete algebraic curve, let ^ be a locally free @B-
Module of rank 2. and let V = P(^). Suppose we have an exact sequence
where 5C\ and .¾ are invertible sheaves on B. Let M2 be the section of V
associated with the surjective homomorphism /?. Show that
a>v/k = 0y{-2M2)®p*{nB/k ® 5?2 ® ^,-1).
(Hint: Let J^ be the defining ideal sheaf of M2. Then we have an exact sequence
o -> ^/^2 - n^ ® ^ - £iM2ik - o.
From this we obtain <M,y/c <E>^, = ^B/k ®-fl^2 if we identify M2 with B. Here
J7/J2'2 = 5C\ ® .¾-1 and wr//c = &V(-2M2) ®p*2' with some JS? e Pic(fi).
Determine J? by making use of the above relation.)
2. Let M and I be the minimal section and a general fiber of the Hirzebruch surface
F„. respectively. For a divisor aM + bl on F„ (a.b > 0). show that the following
equalities hold:
dim H°(F„.& (a M +bl))
_ ( (a + \){b + 1) - \a{a + \)n (b > an).
\ (a+ 1)(6+ 1)-±a(a +l)n {b<an);
dim H{{Fn.@ {a M +bl))
_ ( 0 (b> an).
~ \{a-a){%{a + a + \)-{b + \)} {b < an).
where a = [-1.
(Hint. Use an exact sequence
0 -► &{{a - \)M + bl) -► &{aM + bl) -^<$M{b- an) -► 0.)

Solutions to Problems
Part I
Problems 1.1.
1. S = k[xu... ,xn] - (0) is a multiplicatively closed set, and S' = {1 <g)
/; f € S} is a multiplicatively closed set of k' (¾ ^[xt,..., xn]. Show that (&' (¾
k[xy,...,xn])s, = (k'[xu...,xn])s = k'(xi,...,x„).
2. A necessary and sufficient condition for f{x) to be a separable
polynomial in k'[x] is that k'[x]/(f(x)) is a separable algebraic extension of k'.
Verify that this condition holds by taking it into account that k'[x](f(x)) = k' (¾
k[x]/(f{x)), k[x]/(f(x)) is a separable algebraic extension and k'/k is a purely
inseparable extension.
3. Suppose R is a UFD. Any element £ of the quotient field Q(R) is written as
£ = a/b, where a,b e R and a,ft are coprime. If £ is integral over R, £ satisfies a
relation:
C+CiC~] +•■• + <» =0, c,€i?.
From this derive a relation
a"+ ^,6^-1+ --- + c„fcn =0
and show that b\a.
4. Necessity is easy to show. To show sufficiency, use the ideal I given in the
hint. We have I <£ m by the hypothesis that Rm is a normal ring. We then have
I — R because m is an arbitrary maximal ideal of R.
5. Let p be a prime ideal as in the problem. Since p ^ (0), we may take a
nonzero element f of p. If f = f\fi, we have f\ e p or f2 e p. Hence, we may
assume that f is an irreducible element. Since k[x\,..., x„] is a UFD, / is a prime
element. Namely, (/) is a prime ideal contained in p. Hence, p = (f).
6. (1) Clearly, I Q (x)f](x2,y). Suppose conversely, f € (x) (1 (x2,y). We may
write / = x2g +yh = xl. This expression implies x\h. So / e (x2,x}>). Here (x)
is a prime ideal, and (x2,y) is a primary ideal because \f{x2,y) = (x,y).
(2) Clearly (x) C (/ : x + y). It f £ (I : x + y), we may write (x + y)f =
x2g + xyh. Since yf € (x), we may write f = xf\\ hence, f € (x). It is easy to
show that {x,y) = {I : x).
7. A/m is a finitely generated fc-algebra domain. By Noether's normalization
theorem, A/m contains a polynomial ring R = &[.*],..., xd] over fcasa subring and
A/m is integral over R. Meanwhile, since A/m is a field, so is /? (Lemma 1.10).
Hence, R = k, and A/m is a finite algebraic extension of k.
8. Write f{x) = (x — ai) •••(* — a„) with all a,- € fc. Then A — k[x, (x —
aj)-1, ...,(x — a„)_1], and (x — a\),..., (x -q„) are invertible elements of A. Let
221

222
SOLUTIONS TO PROBLEMS
£/ be the residue class of (x - a,) in A*/k*. We shall show that A*/k* is a free abelian
group with a free basis {£,\ £,„}. If f e A*, then we can write f = g/ Yi"^\ (x ~
aj)x'{gek[x]Aj >0). Noting that/-1 eA*. we may write f ~l = ^/ E["=i(^ -^/)^1
in a similar way. Then gh = n"=i(x - cti)Al+M'■ So g = cY["=i(x ~ a,-)m'(c € fc*,
m, > 0). Thus, we know that A*/k* is generated by ^...., £„. If ^,..., £„ are
linearly dependent, we may assume that there is a relation
Mi + • • • + Ms- = Xv+i4 + i + • • • + y„£„ I 7,- > 0, ]T 7,- > 0 J .
This corresponds to a relation n)-i(x ~~ ai)" = rIl"+i(x ~~ a')" f°r some c € fc*.
This contradicts the hypothesis that k[x] is a UFD.
9. (1) The same as (II.4.9).
(2) Set n = tr. deg^. 2(51). By Noether's normalization theorem, S contains a
polynomial ring R :— k[x\ , xn] as a subring and S is an integral extension of R.
Let n := R nm. Then n is a maximal ideal of i?, and K-dim £„, = K-dimRn. By the
proof of (1.1.19), we know that K-dimi?n = n. Hence, K-dim^4 = tr. deg^ Q(S).
(3) Set r = K-dim/1 and s = K-dim(5/m5). Let [u\, ur} be a system
of parameters of (A.m). Then I = X];=i ufA is a primary ideal belonging to m. So
yfl = m, and hence, \/mB = \JTb. Hence, Spec(B/IB) is the same as Spec(B/mB)
as topological spaces. Thus, K-dim(5/mB) = K-dim(B/IB). Choose elements
wr+i,..., wr+, (u,- G n for each i) of the local ring (B, n) such that {wr+1, ur+s}
(ut := m, (mod/5)) is a system of parameters of the local ring (B/IB,n/IB). Then
n is the radical of J2"i=\ "/-#• % the altitude theorem of Krull, we have htn < r + s.
This entails
K-dim(5) < r + s = K-dim(^) + K-dim{B/mB).
_ 10. Let 91 be the nilradical of A, let A = A/Vl, and let B = B/VIB. Then 5 is a flat
^-module via Tp := f <8>a A: A —> B. (For an /i-module P, we have P ®^B = P ®A B).
Furthermore, K-dim A = K-dim A, K-dim i? = K-dim5. and B/mB = B/mB, where
m = m/91 Thus, we may assume from the beginning that 91 = (0). Let pi,..., p,
exhaust all minimal prime ideals of A. Then p\ U • • • U p, = {a £ A; a is a zero
divisor}, (pj U ••• Up, C {a £ A;a is a. zero divisor} by (1.1.23). Conversely, let
a (^ 0) be a zero divisor of A. Since A is a noetherian ring, there is a maximal element
in the set {an ideal of the form ((0) : b) containing the ideal ((0) : a)}. Let it be
((0) : c). Then it is readily shown that ((0) : c) is a prime ideal. Hence, by (1.1.23),
((0): c) coincides with one of Pi,..., p,. Thus, a € Pi U • • • Up,. Note here that since
v(0) = (0), A has no embedded prime divisors of (0). (The proof is easy.)) We prove
by induction on r := K-dim A that the equality holds in Problem 9 (3). If r — 0,
then A is a field and B = B/mB. So the equality holds. Suppose r > 0. With the
above notations, m ^ Pi U • ■ • Up,. Hence, there exists an element u of m which is not
a zero divisor. Then u is not a zero divisor of B. (The injection w: A —> A, ihui,
induces an injection u <g> B: B —» B, y i—> uy, because B is a flat /1-module.) Hence,
A and B have systems of parameters including u. (Compare (II.6.9) for a proof.)
This implies that K-dim AjuA = K-dim A - 1 and K-dim B juB = K-dim B — 1. Set
B\ := B/uB and A\ := A/uA. By the induction hypothesis, we have
K-dim5i = K-dim/li + K-dim B{/mB\.
Now since B\/mB\ — B/mB, we know that the equality holds in Problem 9 (3).

PARTI
223
11. If M is a finitely generated R-module, it decomposes as M = F © T, where
F is a free R-module and T is a torsion .R-module. If T ^ (0). there exist nonzero
elements a £ R and t G T such that a t = 0. An exact sequence
O^iJ^i?^ R/aR -> 0.
if one applies the functor ©^J. induces an exact sequence
rj-i X U rj-i
T/aT->0.
Thus. M is not a flat
Conversely, if T = (0),
where xa: T —► T is not injective. (In fact, t ^> at = 0.)
^-module if T 7^ (0). Namely, if M is .R-flat. then T = (0).
M is a free i?-module, and therefore, a flat i?-module.
If M is not a finitely generated i?-module, then M is written as a sum M =
lim M; of finitely generated i?-submodules M-K. If M has no torsion elements, then
—*A
each Mi is a free i?-module. Hence. M, as an inductive limit of flat i?-modules, is a
flat R-module. (Note that N ®« (lim Mx) = lim, (N ®R Mx).) Conversely, if M has
torsion elements, then M is not a flat i?-module. This is proved in the same fashion
as for the case of a finitely generated jR-module.
Problems 1.2.
1. (1) For each x e X, there exist an open neighbourhood U of x and the
following commutative exact diagram:
0
0 *&
0
(7 >" ■» U *" <™ I U '
^0
1^)--1^(^1 )(/)
u-/)
where the middle column, which is an exact sequence, exists by the hypothesis that &
is quasicoherent, and where p exists because & is finitely generated. Let {e\ ,e„}
be a free basis of {sf \ u)". Replacing U by a smaller open neighbourhood if necessary,
we may assume that a •/?(?,)(€ T{U,&)) is equal to the image y(£,i) of an element
£,i £ r(C/, (^|(y)^'). Then define rj by e, i-> £,•. For 7 to be injective, we have only
to replace, if necessary, (j/|(y)(/' and 7 by (j/|(y)(/) © (^ |c/)" and (rj, 1), respectively,
where 1 signifies the identity morphism of (
We then have to replace
)(J)
and d by {s/\u){J) <
sequence is exact
)" and (d, 1), respectively. Under this set-up, the following
(^)«©(^)<^Vk)(/)
/)•}'
This implies that X is quasicoherent.

224
SOLUTIONS TO PROBLEMS
(2) For each x e X, there exist an open neighbourhood and the following
commutative exact diagram
0
7
«lr
o—^WuW-^WuY"
s
■is*\u)
(J)
1/2 „
0-
where the second and third rows are the natural exact sequences given by the direct
sum decompositions
/)l
Yand{sf\u)
(J),
\u)i. With U replaced by
a smaller open neighbourhood of x, there exists a homomorphism r: {stf\ u)p —>"§\
such that p = (fi\u)-r. Define a homomorphism ip:
(/)
■&\uby
1i • (vl(j/|t;)(')) = (a\u) • 7 and v\{s*\vy = r -n\. Then <p is surjective. Replacing U by
a smaller one if necessary, we may assume that Kerp = n\{Kgr<p). Furthermore, we
may assume that there exists a homomorphism s: (s^\u)q —► (^|f/)(/) © (-s^lt/)77 such
that Im(s)
\u)W-
(j/|t/)(/)9
CKery) and7ii-5 = a. Define a homomorphism y/: (^Ify)'7' ©(•s^lf/)' —>
|[/)p so that 72-(^1(^)(/)) = 7i -<5 and y/\(^\v)i ^ s-n2. In the above
construction, we employ the exactness of the first and third columns. It is now easy
to show that the middle column is an exact sequence.
2. Answer is given in the problem.
3. Answer is given in the problem.
4. Let Fbe an open set oiX. It suffices to show that a Y{V,stf)-homomorphism
between the modules of Cech rc-cochains
p: C"{1X,F)-
is surjective. Write U = {C/,},g/. For s
•n[/i,,J)-
r(£/,0n.
^r(Fnu,J)
= do in)G/B+1,
r(rn(/(,n-n^,f)
is surjective. (The restriction !F\V of a scattered sheaf & onto an open set U is
a scattered sheaf.) For a = (<*(«))_ve/„+i of C"(V n 11,^), we have only to define
a = (ad)).t€/»+i as follows. If iy = ik for some _/ ^ k, d(s) = 0. If i0 < ii < ■ • • < /„,
take d{s) in r(C/,0 n • • • n Uin,&~) so that d(s) is mapped to 0:(5) by the restriction
morphism. If a is a permutation of {0,1,...,«}, define d(a(s)) = sgn(<7)d(.s), where
a(s) = {iam,...,ia(n]). Then (ad)),e/.+. e C"(U,<F).
5. It is not hard to follow the given steps, one by one. We omit further details.
6. With the notation of Problem 5, H^(X,F) * H^X,^) ^
H"(Y,^\y) = H^(Y,^\y).
7. Let Y be a subset of a noetherian space X, endowed with the induced topology.
Let Fq D F\ D Fi D ■ ■ ■ be a descending chain of closed sets of Y. Let F,- be the
closure of F,- in X. Then F0 D F\ D F2 D ■ ■ ■ is a descending chain of closed sets
of X, and F,- = F, D 7. By hypothesis, F# = F^+i = • • • for some N > 0 which
entails F;v = FN+\ = ■ ■ ■. Hence, Y is also a noetherian space.

PART 1
225
Let U be an open set of X, and let U = {Ua}a€A be an open covering of U.
If \A\ = oo, choose a subset I of A which can be identified with {1,2, 3 }. Let
F„ = U — U"=i ^/ {nGl). Then we obtain a descending chain F\ D F2D Ft,d ■ ■ ■ of
closed sets of t/. Hence, FN = FN+X = ■■■ for some N. If FN ^ 0. (7 g |jf=1 U,- =
[J.Jj 1/,=---. So C/jv+i, C/iv+2 are unnecessary as members of the open covering
11. So we may get rid of UN+\, 6V+2, • • • from U. Then we can take anew UN+l e il
so that (jfLi U/ ^ (J^j (7,-. Since U is a noetherian space, this process ends after
repeated finitely many times. Namely, a finite number of open sets belonging to U
constitute a finite open covering of U.
Let il = {Ua}aeA be a finite open covering of an open set of (7, and let {Fj\
i € 1} be an inductive system of ^/-Modules. Then we know that
H flim^-V(7a) = H lim(^((7a)) = lim J] ^-((/a).
Similarly, we have
J] fum^-Wn (/^) = lim [J #(c/anty).
Since ^- is a sheaf, the exactness of the functor lim entails the exactness of the
following sequence, which corresponds to the axiom of a sheaf:
lim^-W) -► H (lim WW) =t JJ Ajrn^-W D U„).
I ' a€A W ' a.fieA W '
if Wi is a scattered sheaf, then the restriction morphism p'ux: SFj(X) -+ .^-((7) is
surjective. Then
lim//^: Ajm^jW) - Aim^-W)
is surjective. Thus, lim ^ is a scattered sheaf.
Consider the standard scattered resolution of ^),
■£•(#) :0^9-,^ £°m ^ £xm -+ • • • -+ -£"(*-,)
j"
Then {oS1* (.?}); i e /} is naturally an inductive system of complexes. Noting that
(lim 9~i)x — lim 9~-,.x for each x eX,ws can prove by induction that lim S"(9~i) =
^"(lim fj). Hence, lim <Sm (9~,■) is the standard scattered resolution of lim ^. This
—>/ —>/ —►/
entails, by virtue of the result on the coincidence of various cohomologies, that
Hn{X,\\m 9~i) = lim Hn{X,W{).
8. Prove the assertion in each step. It is not so hard.
9. (1) Note that the direct image functor $* (resp., the inverse image functor O*)
with respect to $ = (/, <p): (X, si) -+ (Y, 38) is abbreviated as /» (resp. / *). For an
.^-Module ^, there is a natural homomorphism of .af-Modules y: f */*&' —> ^.
(If (7 is an open set of X, y{U) : lim ^(/ -'(H) -> F(U) is the inductive
limit of the restriction morphisms Pu.f ->{v)-) There is a ^-homomorphism a: "§ ®g$
f*F -+ /*(/ *^ ®^ ^) which corresponds to f *& ®^ y : f *& ®^ f *f^(3r) =
f*(%®<s f*&) -+ f *"§ ®rf & (compare (1.2.9)). In order to show that a is an

226
SOLUTIONS TO PROBLEMS
isomorphism, it suffices to show that ax is an isomorphism for each x e X. Suppose
"S is a locally free ^-Module. Then a being an isomorphism follows from a being
an isomorphism for an open neighbourhood of each point y £ Y. So we may assume
that & is a free ^-Module. We may assume additionally that "§ = 38. Then a is
reduced to the identity morphism l/,sr: f%& —> f*^.
(2) It suffices to show that /' *W ® J" is an injective ^-Module if J" is an
injective ^-Module. Namely, it suffices to show that if tp: & —> 9 is an injective
homomorphism of ^-Modules, tp*: Honv(g\/ *% ® J8") -> Hom^(^./ *r <g> J8")
is surjective. Set £?v := Hom^f. J?). Then we have
Honv(3?./*ir<g> J^) ^Hom^(g,®/*^v.J?)
Homrf^./'r®^) =Hom^(^"®/*^v,J?).
If we note that efv is a locally free .^-Module and ®j// *g'v is an exact functor, the
surjectivity of y* follows from the hypothesis that S is an injective ^-Module.
(3) follows from (1) and (2).
Part II
Problems II.3.
1. Let M be an ./1-module which is not a noetherian module. Then there is an
ascending chain of ^4-submodules of M.
A^o $ NX c ... c Nn c ....
The modules N„ are identified with the ideals (0) © N„ of R. Hence, R is not a
noetherian ring. Meanwhile, (0) © M is contained in the nilradical of R. So SpecP
is the same as Spec A as topological spaces. Since A is a noetherian ring, Spec A is a
noetherian space. Thus, SpecP is a noetherian space, though R is not a noetherian
ring.
2. For p G Spec.4, we have Ip = 0;GA^r(p;)p. Since K{px)p =
(ApJp)APi)p = (0) if p 3> pA and K(p?.)p = K{p;) if p D p;., we have Supp(/) =
3. We omit the solution.
Problems II.4.
1. Since L/K is a finite separable algebraic extension, there is a monic polynomial
f(x) G K[x] such that L = K[x]/(f(x)) with /'(x) ^ 0. Since K' is a splitting
field of f{x) = 0, we may write f(x) = n"=i(x ~~ ai) (each a,- g A^'.« = deg/).
By virtue of the Chinese Remainder Theorem, we then have
(n \ n n
l[(x - a,) * 1[K'[X]/(X - at) = l[K'.
/ = i / /=i / = i
Clearly. Spec(A"' ®K L) consists of n points.
2. Put X = V U F', where F = V = A[.P G V, and P' G F'. Then V xk V'
is an open set of X xk X and (P.F) G V xk V. Since F xt F' = Speck[t.t']
and (Ax//t)n(F x^ F') = Specifc[M']/(f - O, we have {P,P') g Ax//t. Similarly,
(P\ P) G AY//t. In order to show that Ax/k = Ax/k U {(P\ P), (P, P')}, we have only
to note that X xkX = {V xk V) U (F xk V) U (F' x^ F) U (F' x^ V) and look
at the closure of AX/k restricted on respective open sets.

PART II
227
3. (1) Let X' = X xY Y'. To show that the base change /': X' -> Y' is
a closed immersion, we may replace Y' by an affine open neighbourhood V of a
point of f'(X'). Furthermore, we may assume that U' Cg~l(U) for an affine open
set U of Y. Set U' = Spec ^4' and U = Spec A. Since f is a closed immersion,
we may write f ~X{U) ~ Spec5. where B = A/I, I being an ideal of A. Then
/'-!([/') = SpecU' ®A B) = Spec{A'/IA'). Hence, /': X' -> 7' is a closed
immersion.
(2) We may assume Z = Spec^. Then 7 = SpecB. where B = A/I with
an ideal I of A. Similarly, X = SpecC, where C = B/J with an ideal / of B. Let
n: A —> 5 be the natural residue homomorphism, and let J = n~x (J). Then / C /
and / = ///. Hence, C = ./1//. Namely, we see that g-f: X —> Z is a closed
immersion.
4. (1) Set 7 = y/x. Let A be an algebraic closure of A, let a = a1/", and let ew be a
primitive rcthroot of 1. Then t" —a splits as t" -a = (t — a)(t -aco) •■■{t — acon~l)
in k. Hence, if a1/"7 ¢ A (for each w, 1 < m < n.m\n). then t" — a is irreducible
over A. Thus, j" - ax" is irreducible over & under the same condition as above.
Since y" - ax" = Y\"Z0 (y - aco'x) over k. C (¾ k is a union of « lines y = aro'x
(0 < i < n) through the point of origin on A|.
(2) Argue as in (1) above.
5. (1) This part is clear.
(2) Let & be a discrete valuation ring of k(Y) containing k. Then t G & or
u G <f. If f G <f, then <f = it[?](,_„) for some a £ k. If w G &'. then <f = k[u\u_fi)
for some /? G k. If /? ^ 0, then <? = Ar[?](,_/? L).
(3) Let & be a discrete valuation ring such that A c ^f and k(7) c Q{@). Then
(^ D A(7) is a discrete valuation ring of k(Y). i.e., the quotient field of & n A(7)
is A(7). So <f n A(7) = Jt[?](,_a) (a G A) or & n A(7) = k[u\u). Namely, the
specialization (of the generic point) of Y along & is either t = a or t ~ oo, and it
is thereby uniquely determined. Hence, by (II.4.15). Y is separated over k.
(4) Let & be a discrete valuation ring such that A: C <f and A(7) c Q(@).
Then, as in (3), & dominates a point f = a or / = oo. Hence, 7 is complete.
(5) There is no point on X which is dominated by the discrete valuation ring
MM1(«) (' = w_1) of A(Z) = k{t). Moreover. k[t]^ dominates two points P and P'.
6. An answer is given in the problem.
7. Let (9 be a discrete valuation ring of A(A") containing A. Since X is complete,
there exists a closed point x of X such that <f > &x.x- Note that (f^.x is a normal
local ring of dimension 1, hence, a discrete valuation ring (1.1.29). Then & — &%.x-
Since X is separated over k, such a point x is uniquely determined by &. Thus, we
have an injective mapping of sets
ip: {&; a DVR of k{X),k c 0}-> {x; a closed point of X},
<p{&) = x if and only if & = ^fx..v.
Clearly, ip is surjective, too.
8. Put X ~ Spec ^4 and S — Spec jR. The morphism /: X -+ 5* is a quasicompact
morphism. (We have to show that f ~1{V) is quasicompact for every quasicompact
open set V of S. V has a finite affine open covering V = |J"=1 £>(/,-), where /,- G i?
and / ~l(D(fi)) = D(tp(fj)),ip: R ^> A being the ring homomorphism associated
with /. Since £)(^(/,-)) is a compact open set of X and / _1(^) = U"=i ^(v (//))-
/ _1(F) is also a compact open set.) Thus, / being a finitely generated morphism

228
SOLUTIONS TO PROBLEMS
is the same as / being a locally, finitely generated morphism. Namely, for each
x e X, there exist affine open neighbourhoods U of x and V of s = f{x) such that
U C f -1{V) and T{U,@u) is a finitely generated T(F,<fV)-algebra. If D{b) is an
affine open neighbourhood of s in V, we may replace V, U by D{b), f ~l{D{b)) n
C/. respectively. Thus, we may assume from the beginning that V = £)(6). Then
Y{U,@u) is a finitely generated i?[6_1]-algebra. So r(C/,<ff/) is a finitely generated
.R-algebra. After all, there exists a finite affine open covering X = (J"= { U/ of X such
that Aj := T(Uj.@x) is a finitely generated .R-algebra for each i, 1 < i <n. If £>(g)
is an affine open set contained in Uh then ^4[g_1] is a finitely generated ^4-algebra.
Hence, ^4[g_1] is a finitely generated ,4,-algebra. This entails that ^[g_1] is a finitely
generated i?-algebra. This remark allows us to assume that C/, = D{gt) {gj eA). Let
{"ti/s?1' 1^7^ N} (u/j G A, M > 0) be a system of generators of the i?-algebra
A\gf1]. Taking utj repeatedly if necessary, we may assume that N is independent
of i. On the other hand, since X = (J"=1 £>(#,)> there is a relation X)"=1g,/!/ = 1
{h; e A). Then we claim that A = R[ujj:,gt, h;; 1 < i < «, 1 < ;' < iV]. In fact, for
each £ £ A, there are F,- G /?[g,-, w,y; 1 < ;' < N] and an r > 0 such that g\^ = Ft
for each i, 1 < / < n. We have also
/ n \rn n
1 = Y,gih, ) = J]*[#,, Hi g Kfe,, A/: 1 < i < «].
Hence, £ = E?=i gf#/S = E?=i ^ e R[uij,glM 1 <»"<«, 1 < i < «].
9. It is not hard to follow the given steps, one by one.
Problems II.5.
1. (1) This is clear by the definition.
(2) Let x G VI. Write x = xo + x\ -\ as a sum of homogeneous elements.
Suppose Xj G VI {i < r) and xr ¢ VI- Since xN GI for some N > 0, (x^)^ = x^ +y
with j G (x0,... ,xr_!)rAf C v^- Since {xN)rN G /, it follows that x/^ G VI- So
xr G \/7. This is absurd.
2. Repeat the arguments in (Part I, Chapter 1), replacing all ideals and elements
there by homogeneous ideals and homogeneous elements. We omit the details of the
proof.
3. A sketch of solution is given in the problem.
4. Let QJ = {Fa}xga be an affine open covering of Y such that {f~l{V\);
X g A} is an affine open covering of X. Set Ux = f~l(V{). Clearly, two Cech
complexes C (93, /*&) and C* (11,^) are isomorphic, where U = {Ux}XeA. Hence,
HP(93, /„?") = £f (2T, ^) for each/> > 0. Note that for each m > 0, VXi n • • • n KAm
is an affine open set of Y and 1¾ n • • • n Ukm is an affine open set of X. By (II.3.14),
H"{VXln---n VXm,f,9-) = Hn{Uh n • • -Vi t/^,^) = (0) for each « > 0. Then
by virtue of (1.2.22), /^(33,/,,^) = Hp{YJ^) and //^(11,50 ^ HP{X,^) for
each/> > 0. Thus, HP{X,3r) = HP{YJ%^) for eachp > 0.
5. CI(R) is the quotient group of the free abelian group ©htp=1 Z[p] generated
by all height 1 prime ideals of R modulo linear equivalence. If R is a UFD, then
every height 1 prime ideal of R is a principal ideal (compare Problem 1.1.4). Hence,
C1(jR) = (0). Conversely, suppose Cl{R) = (0). Let p be a height 1 prime ideal of
R. Then P = (/ ) (/ G Q{R)) by hypothesis. We claim that / G R. In fact, f & Rq
for every height 1 prime ideal q of R. [f G pRp if q = p, and / G (i?q)* if q ^ p.]
Hence, / G Hq ^q =R (LI.33).

PART 11
229
We shall show that any irreducible element a of R is a prime element. In fact,
any prime divisor p of a principal ideal aR has height 1 (1.1.33). Since p = (f ) with
f G R, we have f\a. Since a is an irreducible element, we have aR = fR. Namely,
a is a prime element. This implies that R is a UFD.
6. We may assume that the projection n: ~PN —> P" is given by
(xo,xi,...,xn) i—> (xq,xi,...,x„), where (xq.x\ ,x^) are homogeneous
coordinates of P^. Define a linear subspace L by xo = ■ ■ ■ = x„ = 0. Then P^ — L is
the domain of definition of n. Put Y = PN - L, Z = P" and p = n\ y. p is an affine
morphism. Since ip = n • y/ is a morphism, \//(X) n L = 0. For each x e A", set
3> = y/(x) and z = <p{x). By hypothesis, there exists an affine open neighbourhood
W = Spec/? of z such that ip~l(W) = Spec(i?//), where / is an ideal of R. Putting
p~l(W) = Spec/? and ip~l{W) = Spec A, we obtain the following commutative
diagram
R —^ A £ #//
where/? is identified with a subring of B. We have "/? = y/|v-i(^) and "a = y>| -i^.
Apparently, /? is a surjective ring homomorphism. Hence, A = B/I for an ideal /
of B. Thus, ^ is a closed immersion.
7. It is not hard to verify the assertions given in the steps.
Problems II.6.
1. Put A = R[x\,...,x„] and £1 = Q.a/r. For an arbitrary A -module M,
the ^4-homomorphism ip: Hom^(Q,M) —> Der^(^4,M) is an isomorphism, where
tp(a)(a) =a(da) (a £A) for a eHom^(fl, M). Besides, Q. is an A -module generated
by dx\, dxn. Any element 8 e Der/?(^4. M) is given by assigning elements 8{xt)
of M for each i, I < i <n which can be arbitrary elements of M. If a\dx\ -\ +
a„dx„ = 0 (a,- G .4) is a nontrivial relation, then we have ai<S(xi) + 1- a„d(x„) = 0.
This is a contradiction because we can taked(x\),...,d{x„) arbitrarily. Hence, Q. is
a free ^4-module with a free basis {c/xi,..., dx„}.
2. Let yf be the sheaf of nilradicals of X. Since X is an algebraic A:-scheme,
jVr = (0) for a positive integer r. We may assume that JVr~l ^= (0). Put J? = Jfr~x
and 7 = (X,$x/<-?)■ We have an exact sequence (II.6.18):
If fix/A- = (0), then Qy/k = (0)- Next we repeat the above argument for the sheaf of
nilradicals of Y. We thus know that £ixIcd/k = (0). In order to verify the requirement,
we may assume that X = XKd. Next let X\ be an irreducible component of X.
Applying (II.6.18 (3)) to the natural closed immersion f: X\ —> X, we obtain an
exact sequence
f *Qx/k —> Qxi/k —> Q-Xi/x —> 0.
Here it is readily shown that £lxx/x = (0)- (To show this, we may assume
that X\ = Spec/?, X = Spec.4, and B = A/I. By means of an isomorphism
WomB{Q.B/A,M) = DerA(B,M), we have only to show that any element of
DerA(B, M) is zero.) Then £lx,/k = (0). Noting that dim X = sup, dim X-, when X,
ranges over all irreducible components of X, we have only to prove that dim X = 0
in the case where X is irreducible. We assume that X is reduced and irreducible. Set

230
SOLUTIONS TO PROBLEMS
K = k(X) and apply (11.6.18(3)) to the natural morphism rj: Spec A -> X. We
know as above that £lspec(K)/k = (0)- Hence, Clfc/k = (0). Let n = tr.deg^ A, and
let {x\ xn} be a transcendence basis of K over k. Setting Kq = k(x\,....x„),
we have an exact sequence
Hence, Q.k/k0 = (0)- This implies, as shown in (II.6.19), that A" is a separable
algebraic extension of K0. Namely, {x\ — ,xn} is a separating transcendence basis
of K over k. Let M be a A-module. Then Derk(K, M) = Der^Ao, M). (Write A =
Kq(9), and let /(/) be a minimal polynomial of 9. Let <S G Derfc(A"0, M). Then <5 is
uniquely extended to an element of Der^(K, M) by putting 6(6) = -f '(9)~lf3(9),
where fs(t) is the polynomial obtained by applying 5 to the coefficients of /(0-)
Since Der^Ab, M) = UomKo(QKo/k,M) = Hom^0(©;=1 A0 <**,, M) ^ M©" (= a
direct sum of n copies of M) and Y>erk(K,M) = Hom/^Q^.M) = (0), we have
n = 0.
3. Let P G Sing F. Suppose Z0 ^ 0 at P. Let x,- = Xj/X0 (1 < i < n) and
/(xi,. ..,*„) = (^)-^(¾...., *„), where d = degP. By (II.6.15), P e Sing F if
and only if f(P) = (df/dxi)(P) = 0 for each i, 1 < i < n. (Note that dim V = n - 1.)
Noting that df/dxi = Uo)~(J_,)(dF/dX,•), we know that P G Sing F if and only
if F = (dF/dXi) = ■■■ = (dF/dXn) = 0 at P.
Make the same argument in the case Xj ^= 0 at P.
4. By hypothesis ClX/r = (0)- This entails 0.Xk/k = (0), where K = k(Y) and
Zjf=Ixr Spec K; XK is a reduced and irreducible algebraic A-scheme. Let L =
k(X). Then L = K(XK). Since £1Xk/k = (0), we have QL/K = (0). So, tr. deg^ L = 0
by Problem II.6.2. It then follows that dim A" - dim Y = tr.deg^ L = 0.
Part III
Problems III.7.
1. (1) There exists a 3 x 3 symmetric matrix A with entries in k such that
F(X0,XuX2) = (^0,^1,^2)^(^0,^1,¾). Since the characteristic of A: is not 2,
there exists an invertible matrix P such that PA'P is a diagonal matrix. Hence,
we may assume from the beginning that A is a diagonal matrix, i.e., F = a^X^ +
a\X\ + a2X22. According to rank A, further changes of variables reduce F to one
of the standard forms (i), (ii), and (iii).
(2) C is irreducible and reduced only in the case (i). Then C is the image
of $|2/>|: P1 —» P2, where P is a closed point of P1. (If we choose homogeneous
coordinates (£0,£i) of P1 so that P is given as (1,0), then we have H°(Pl,@(IP)) =
k$ + fc&f 1 + m Set X0 = $, Xi = tf and X2 = &fi. Then C is given by
XqXi = X22 and ¢^1 gives rise to an isomorphism between P1 and C.)
(3) As in (1), we can express F — a^X^ + ci\X2 + a2X22 + a^Xf, where a,- = 0
or 1 and a0 > a\ > a2 > «3. Hence, after a suitable change of variables, it is classified
into the following standard forms: (i) X0Xi = X2Xi, (ii) X0X2 = X\, (iii) X^XX = 0,
and (iv) Xq = 0. In the cases (ii), (iii), and (iv), the quadratic hypersurface is a
projective cone consisting of lines which pass through points of a conic on P2 and
the point (0,0,0,1). In case (i), it is isomorphic to P1 x P1. (An isomorphism is
given by (t0J\) x (ho, «1) >-► (touo,t\ui,t0u\J\u0).)

PART III
231
Problems III.8.
1. Set & = &v.p and m = mv,p. Since P is a nonsingular point of V, m/m2 =
ku + kv for a local system of parameters {u,v) at P, where u = u (modm2) and
v = v (modm2). If we take an affine open neighborhood Spec A of P, we have
(<f.m) = {A <xn. 9JL4 OT), where 9JT is a maximal ideal of A. Now for (a,/?) £ A:2 -(0.0),
choose g e m such that au + flv = g (modm2). Since geia. write g — a/b with
a,b eA and ft ¢ 9Jt. Let Z> be a closed subset of Spec .4 defined by a = 0. Then each
irreducible component of Z> has dimension 1. (Each prime divisor of a A in a normal
ring A has height 1.) Furthermore, there is a unique irreducible component of D
passing through P, and P is a nonsingular point on the irreducible component. Let B
be the closure of the irreducible component passing through P. which is an irreducible
algebraic curve. B is nonsingular at the point P. Suppose the irreducible algebraic
curve C is defined by f = 0 in an open neighbourhood of P, where f € mVP. Then
we have
i{C.B,P) = lengths/ (/, g) = length k[[u,v]]/ (/„ + •• • ,(aw+/?«) + •••)■
Consider the case /? ^= 0. Then /t[[w, u]]/(g) = k[[u]] and u = Am + (terms in u of
degree > 2). where A = —a/p. Choose (a,/?) so that /^(1, A) ^ 0. Then we have
■ i^ n r>\ i *u / rr n / / r /• i ia u , / terms in w of degree \
i(C,B;P) = lengthk[[u]] / 1/^(1.1)^ + ( > / + ^ J
= lengthfctMJAu") =//.
Namely, we have,u(C,P) = MPeB i(C.B;P).
2. As given in the Hint, coC/k — @vAn ~ 3)|c- Since C ~ «// on P2 with a
hyperplane i/ on P2, there is an exact sequence
0 -► <9vi{-n) -+ <9vi -+ &c -+ 0.
Taking tensor products with the invertible sheaf @vi (n — 3), we have an exact sequence
0-xfP2(-3) -x?P2(n-3) -^<fp2(« -3)(gxfc ^0.
Since /;'(P2,<fp2(-3)) = 0 for i = 0,1 (II.5.12), the cohomology sequence entails
ho{C.<?P2{n-3)®0c) = h°{P2,@P2{n-3)) = (""') = («-1)(«-2)/2. Namely,
we have g = (« - 1)(« - 2)/2.
3. As explained in the Hint, we have for f e k(V):
f £ M iff f £ M(L) (= the module associated with L)
and all the coefficients of f, viewed as an
element of k[[u, v]]. vanish for the terms of degree
between 0 and //-1.
Let n = dimfc M(L), and let {f\,..., /„} be a /obasis of M(L). Then we have
(*) f = a\f\ H + a„f„ all a,-G A:.
Since we can choose f\,...,f„ from <9v.p, the comparison of the coefficients of terms
of degree between 0 and ju - 1 in the relation (*) yields altogether /i(ju+1)/2 linear
equations in a\,...,a„. Hence, we have
dmu M > n - ^-—-.

232
SOLUTIONS TO PROBLEMS
Problems III.9.
1. (i) A is a field if it is reduced. Since the characteristic of k is zero and
dim^ A < oo, A/k is a finite separable algebraic extension. By the result verified
in the proof of (II.6.19), we have £lA/k = (0)-
(ii) Conversely, suppose QA/k = (0). Suppose A is not reduced. Let m be the
nilradical, and let A — A/m2. Then by an exact sequence
J/J2 -> SlA/k ®AA^ ^A/k ^0 (J =m2),
&A/k 7^ (0) provided 0.A,k ^ (0). So we may assume from the beginning that m2 = (0).
(iii) Define a ring homomorphism ip: A —* A by ip(a) = a ® I. Since A is /oflat,
ip is an injection. So A is viewed as a subring of A. If each At is reduced, then ^4
is apparently reduced which implies that A is reduced.
(iv) Since fi 7^. = fi^ (¾ k = I~[/=i ^,/^ we nave tne equivalences: Q^ = (0)
iff Qr,£ = (0) iff QA,/~k = (0) for each i. Hence, we may assume k' = k, and we
have only to show that "Q^ = (0) implies m = (0)". Then A decomposes as
A = k © m and the product is given by (a.x) ■ ifi,y) = (o/?,/?x + ay). If {a,x)
corresponds to a e A, then a = a (modm) and x = a - a. Set x = d(a). If (/?, j)
corresponds to Z> G ,4, then 8{ab) = fix + ay = /?<S(a) + aS(b) = bd{a) + ad{b).
Thus, 6 is a /oderivation of A with values in m. Suppose m ^ (0). Then 5 ^ 0. So
Derfr(v4,m) = Hom^Q^.m) ^ (0). This contradicts the hypothesis £lA/k — (0).
2. We may assume that X is connected. Let X — X\ U • • ■ U Xr be the irreducible
decomposition of X. If A" is reducible, there are two distinct irreducible components
Xj.Xj of X with Xj n .X) 7^= 0. Suppose 1^¾^ 0. Let P be a closed point of
X\ C\X2, and let <f = @x.p- Then A'i and Xi correspond to minimal prime divisors
pi,p2 of(f. By (1.1.23). pi = ((0) :a\) and p2 = ((0) : a2) for some aua2 £&. Since
(f is an integral domain, pi = p2 = (0) which is a contradiction. What is necessary
in this proof is that every local ring of X is an integral domain.
3. Let M be the k-module associated with A, which we take as a submodule of
H°{V,<?v{Di)). Let (Ti,(T2 be elements of M such that D, = (<7,-)0 (i = 1,2). Then
any element a of M is uniquely written as a = a\o\ + a2a2 with 0:1,0:2 G &. If
P G SuppA n Supp£>2, then o\{P) = o2(P) = 0. So a{P) = 0 and P e Supp((7)0.
Namely, P G BsA. The inclusion BsA C SuppDi n Supply is clear.
4. It is not hard to solve problems following the instructions in the steps. We
omit the details.
Problems 111.10.
1. (1) Set x — X/Z and y = Y/Z. Then y2 = (x — a\) • ■ ■ {x — a„) is the
defining equation of Q := C n A2, where A2 is the complement of the hyperplane
if : Z = 0 in P2. Since C C\ H = {(0,1,0)}, we have only to show that C0 is
irreducible. Set f = y2 - ri/=i(x ~~ a>) an(^ R = k[x,y]/(f ). If we view f as a
polynomial in y over the polynomial ring k[x], then f is irreducible by Eisenstein
criterion of irreducibility. Since &[x,y] is a UFD, f is a prime element. Hence, R
is an integral domain. Thus, C is an irreducible algebraic curve.
(2) Applying the Jacobian criterion in the Problem II. 6.3, we know that C is
nonsingular if n = 3 and that Poo := (0,1,0) is the unique singular point of C if

PART III
233
n > 3. Set u = X/Y and v = Z/Y. In an open neighbourhood Uq = Spec/cfa.t;] of
Px in P2, C is then defined by an equation v"~2 = (u — a\v) ---(11- a„v). Thus,
//(C, Poo) —n-2. Let a\: U\ —» 1¾ be the monoidal transformation with center at
the point Px. let £, = a~x(P00) and let Ci = <r((C). Then t/, = 1½ U £/n with
Ui0 = SpecA:[w,wi] and U\\ = Specfc[Mi,i;], where ui = v/u and «i = u/v. The
curve Ci does not meet the line v = 0 on U\ 1. (The defining equation of C\C\U\\
is 1 = v2(«i — a{)- ■■ («i - a„).) Hence, Q C t/10, and C is denned by an equation
v"~2 = u2(\ -aiv\)---(1 - anv\). Its singular point is P^ : (u,v\) = (0.0) and
fi(C\,P^) = 2. Let 02: t/2 -* U\ be the monoidal transformation with center at
the point P^, and let C2 := a'2(C\). If we set u = v\u2, Ci has a unique singular
point P®: (u2, vx) = (0,0) on U2\ = SpecA:[w2, i>i], and C2 is defined locally at P&
by u\ = v"~4 • (an invertible element). So fi(C2, P$) = 2. In this way, we can show
that C has infinitely near points PQ P%}(r = [f]~ 1) and nas multiplicity 2 at
these points.
(3) We compute
Pa(C) =l-(C + KP2-C) + l = ~n(n - 3) + 1 = i(n - l)(n - 2),
and
g(C) = Pa(C) - ~(n - 2)(n - 3) - r = (n - 2) - (g] - l)
-1 (cf (111.10.4)).
= n —
2. (1) implies 2. Use the arithmetic genus formula and the result pa(C) > 0.
Since -2 <pa(C) -2 = (C1) + (C ■ Kv) < -2, it follows that (C2) = (C ■ Kv) = -1.
(2) implies 3. Use the arithmetic genus formula.
(3) implies (4). By Lemma 10.4, C has no singular points, and g(C) = 0. So
C =^ P1 by Theorem 7.6.
(4) implies (1). Use the arithmetic genus formula.
Problems III.ll.
1. Put Ej = o-x(Pj). By (III.8.8) and (III.10.3), a*C = C + Ylt=\miEi and
Kv ~ a*(Kpi) + Yl\=\ Ei- Hence, we have
^ = (a*C)2 = (C'2) + £m2,
/ = i
r r
(C'-Kv) = (o*C-<j*(KPi)) + Y,m: = -3^ + ^m,-.
/=1 /=i
2. The condition for C" to be a (-1) curve is (C2) = (C • Kv) = -1. This is
the same as
/- /■
d2 -^mj = -3d +^m, = -1.
/=1 /=1
Suppose mx > • • • > mr. Setting mr+1 = ■ • • = m8 = 0 if necessary, we may assume that
a is the monoidal transformation with centers as 8 points. (The condition mr+i =
• • • = m8 = 0 is equivalent to the condition that C does not pass through Pr+\, ■ ■ ■, Pg-

234
SOLUTIONS TO PROBLEMS
The proper transform of C by the monoidal transformation with centers at 8 points
has the same self-intersection number as the proper transform of C by the monoidal
transformation with centers at r points.) Eliminating d from the above two equations
in d and m, we obtain an equation
8
18 = ]T(m; - 1)2 + Y^m, - mj)1.
i' = l <'</
3. Consider the case r = 8. The other cases are treated in the same way. First of
all, there are 8 exceptional curves of the monoidal transformation a with centers at
8 points, which are (—1) curves. We next look for (—1) curves which are the proper
transforms by a of certain irreducible curves C on P2. We follow the classification of
all possible cases in the Problem 2. In the case of (1; l2), C is a line passing through
two points among P\,...,P%. Hence there are (*) of such C's. In the case of (2; l5),
C is a conic passing through 5 points among P\...., Pg. Hence, there are (®) of
such C's. In the case of (3; 2, l6), C is a cubic curve passing through 7 points of
P\ , P%, one of which is a double point of C. So there are (®) • (J) of such C's.
Similar observations entail that there are altogether as many (—1) curves as
KKHKMKK)-
Problems 111.12.
4. Let V = F{&~), let p: V —> B be the structure morphism, and let Vn be
the generic fiber of/?. Then Vn = PL . and Qvjk^ = cov/k <g> k(rj). So we can
write eov/k ^ <f(-2A/2) <g> p*S? with & e Pic(B) (III.12.8). Meanwhile, D.B/k <g>
J"/J"2 = cov/k®&M2 (Hint) and J"/^2 = SS\ ® ^x. (Write down local denning
equations.) Hence wv/k <g> &Ml - (<M-2M2) <& 0Ml) ® % - ^\ ® -¾-2 ® % and
cov/k ® @m1 - ClB/k ®%\®&2X- So we obtain & = QB/k <g> S?2 ® -S?,-1.
2. Taking tensor products of &{aM + bl) with each term of an exact sequence
0 -> &(-M) -> <? -> &M -» 0.
we obtain the following exact sequence
(*) 0 -> <f((a - 1)M + bl) -» ^(oM + 6/) -► ^Pi (6 - an) -> 0.
Suppose b > an and a > 1. We proceed by induction on a. Since b > (a — \)n, we
assume that
h°(Fn,0((a - l)M+bl)) = a(b +1)- ha - l)an
^(^,^((^-1^ +6/)) =0.
The cohomology sequence attached to the exact sequence (*) entails
h°{F,@{aM + bl)) = a(b + 1)- -{a - \)an + (b-an+l)
= (a + l){b + l)--a{a + \)n,
hl(F„,@(aM + bl)) =0.

PART 111
235
In order to show that h°{F„.tf{M + bl)) = 2{b + 1)-/7, proceed by induction on
b by making use of an exact sequence
0 -> <9{M + {b- 1)/) -* &{M + bl) -» ffi -f 0.
The other cases can be treated in a similar fashion.

List of Notation
A := B
A-B
(I : a)
V7
Rs-Rp
K-dimi?.dimi?
&>&'
ht(p)
tr.degkK
Ga\{K/k)
lim Aj.lim A,-
tfu].stn
Hom^.^)
Keryj.Cokeryj.
Imy>
Un{X.9-)
H"(X.9-)
R"f*9-
Spec(i?)
give the set B the name A: define A by the condition B.
A\{A n 5) if A and B are subsets of a set S: difference of A
and B if they are elements of an abelian group.
ideal quotient; (I : a) = {x e R: ax £ 1} if I is an ideal of
a ring R.
radical of an ideal I.
quotient ring, localization.
Krull dimension of R.
a local ring (<f.m) dominates a local ring (<f".m')- i.e.. <9 D
<f".mn<f" = m'.
height of a prime ideal p.
transcendence degree of an extension field K/k.
Galois group of a normal algebraic extension.
direct product of sets. etc.
direct sum of sets. etc.
inductive limit, projective limit.
direct sum of sheaves of modules.
direct sum of |/| copies (or n copies) of a sheaf of commutative
rings.
tensor product of j/-Modules.
group of homomorphisms of j/-Modules.
j/-Module of homomorphisms of j/-Modules.
dual jZ-Module. /o)firf(^.i).
kernel, cokernel. and image of an j/-homomorphism
if : & -> § of j/-Modules.
direct image, inverse image of a module of sheaves.
module of sections of a sheaf of modules !? over an open set
U.
cohomology group defined by an injective resolution of a sheaf
of modules 9~.
Cech cohomology group of 37 defined by an open covering it.
Cech cohomology group of &; by the definition, it is
lim #"(11.50-
<—u
higher direct image of &.
spectrum of a commutative ring R.
237

238
LIST OF NOTATION
V(I).D(I).D{a) closed set. open sets in the Zariski topology.
R. M quasicoherent sheaves associated with R and an /^-module M.
p.v prime ideal of R corresponding to a point x of Spec(i?).
k{x) residue field of a point x.
Aj? w-dimensional affine space over k.
dim X Krull dimension of a scheme X.
codim^Z dodimension of a closed set Z in X.
XKi reduced form of X.
X{k) set of ^-rational points of a ^-scheme X.
k(X) rational function field of a ^-scheme X.
U Xj direct sum of schemes.
X xsY fiber product.
f : X —> Y a morphism of schemes.
Homs(X. F) set of S-morphisms from an S-scheme X to an S-scheme Y.
Xv fiber of / : X —> Y over a point y of Y.
f : X ■ ■ ■ —> Y a rational mapping of schemes.
Vf graph of a morphism (or a rational mapping) f.
x —> x' specialization of points x to x' on a scheme.
A+ irrelevant ideal of a graded ring A = X^«>o^«-
Proj^ homogeneous spectrum of A.
V+{E).S+(E). colsed set. open sets of ProjA
D+{a)
&x(i) A(e)~.
P;" «-dimensional projective space over k.
S'(9") symmetric algebra of a locally free Module.
P(^) projective bundle associated with &.
Pic(A') Picard group of X.
D\ ~ Z>2 linear equivalence of divisors.
D\ « Di algebraic equivalence of divisors.
D\ = Di numerical equivalence of divisors.
C1(X) divisor class group of X.
D > 0 an effective divisor.
SuppD support of a divisor.
(/) divisor of a rational function /.
(/)o- (/)00 zero part, polar part of /.
\D\ a complete linear system.
M{L) ^-module associated with a linear system L.
$>l rational mapping attached to a linear system L.
<P^f rational mapping defined by H°(X. SC) for an invertible sheaf
21 on X.
BsL set of base points of a linear system of L.
\D\ - Yl"=[ miPi linear subsystem of \D\ consisting of divisors on an algebraic
surface V which pass through the points P,■ (1 < i < n) with
multiplicity w,-.
X • //(a) pullback of a hyperplane H(a) by ¢^.

LIST OF NOTATION
239
R completion of R with respect to a linear topology.
R\xx x„| formal power series ring in n variables over R.
£Ia/b-®-x/y module of differential forms, sheaf of differential forms.
Der#(,4. M) ^-module of 5-derivations of A.
SingX set of singular points of an algebraic variety X.
^x/k tangent sheaf of X.
jVy/x normal sheaf of a closed subvariety Y of X.
ojx/k canonical sheaf of X.
Kx canonical divisor of X.
ti{X.&) dim/,//'(X SO-
x{X.&~) Euler-Poincare characteristic of W.
g{C) (geometric)genus of an algebraic curve C.
p(C) arithmetic (virtual) genus of C.
(^1-^2)-(^11¾) intersection number of invertible sheaves (or divisors) on an
algebraic surface V.
{SC2). {D2) self-intersection number.
i{D\.D2;P) local intersection multiplicity at a point P.
/u(C;P) multiplicity of an algebraic curve C on V at a point P.
pa{D) arithmetic genus of a divisor D.
P„.P„{V) M-genus of an algebraic surface V.
Pg-Pz(V) geometric genus of V.
q{V) irregularity of V.
p(V) Picard number of V.
NS(K) Neron-Severi group.
F„ Hirzebruch surface of degree n.
elm/> elementary transformation with center at a point P.

Bibliography
1. R. Godement. Theorie de faisceaux. Hermann. Paris.
2. A. Grothendieck. Sur quelques points d'algebre homologiques. Tohoku Math. J. 9 (1957). 119-221.
3. A. Grothendieck et J. Dieudonne. Elements de geometrie ulgebrique (EGA). Publication de I'lnst.
Hautes Etudes Sci. Publ. Math.. 4, 8, 11, 17, 20, 24, 28, 32.
4. R. Hartshorne. Algebraic geometry. Springer-Verlag. Berlin-Heidelberg-New York.
5. H. Matsumura. Commutative algebra. Benjamin. New York. 1970.
6. Commutative ring theory. Cambridge Univ. Press. London.
7. M. Nagata. Theory of commutative fields. Translations of Mathematical Monographs, vol. 125. Amer.
Math. Soc. Providence, R.I.. 1993.
8. Local rings. Interscience Tracts in Pure and Applied Mathematics. John Wiley & Sons. New
York. 1962.
241

Index
(-1) curve. 199
Abelian presheaf. 27
Abelian sheaf. 29
Absolutely minimal model. 208
Absolutely normal ring. 184
Affine algebraic scheme. 74
Affine algebraic set. 73
Affine algebraic variety. 74
Affine morphism. 96
Affine open set. 81
Affine scheme. 61
Affine space. 73
Algebraic curve. 103
Algebraic scheme. 89
Algebraic subvariety. 89
Algebraic surface. 152
Algebraic variety. 89
Algebraically equivalent. 187
Algebraically independent. 3
Altitude theorem of Krull. 24. 139
Ample. 127
Arithmetic genus. 160. 192
Arithmetic genus formula. 192
Ascending chain condition. 9
Associated fc-module. 124
Base change, 87
Base point. 125
Birational invariant. 197. 208
Birational mapping, 100
Blowing-up. 169
C,-field. 215
Canonical divisor. 151
Canonical sheaf. 150. 151
Cartier divisor. 123
Cauchy sequence. 133
Cech cochain. 44
Cech cohomology. 46
Cech H-coboundaries. 45
Cech H-cocycles. 45
Cech resolution. 46
Closed embedding. 81
Closed fiber. 184
Closed immersion. 81
Closed morphism. 93
Closed subscheme. 81
Codimension. 64
Cofinal. 133
Coherent. 33
Cohomology. 41
Cohomology functor. 43
Cohomology theory. 43
Cokernel. 29
Compatible. 26
Complete algebraic variety. 92
Complete linear system. 124
Completion. 135
Composed of a pencil. 180
Conormal sheaf. 150
Contraction of the ( — 1) curve. 201
Converge. 134
Coordinate neighborhood. 152
Coordinate ring. 61. 73
Coordinates. 73
Cross normally. 168
Cross section. 188. 212
Cup product. 154
Defining ideal. 73
Degree. 111. 159
Degree n homomorphism. 113
del Pezzo surface. 210
Derivation. 143
Diagonal morphism. 88
Differential module. 144
Differential sheaf. 150
Dimension. 11
Dimension of linear system. 124
Direct image. 30. 32. 206
Direct sum. 83
Discrete valuation. 16
Discrete valuation ring. 16
Divisor. 121
Divisor class group. 122
Domain of definition. 99
Dominant. 93
Dominant rational mapping. 100
Double complex. 50
Dual projective space. 112
Dual sheaf. 32
243

244
DVR. 16
Effective divisor. 121
Elementary transformation. 219
Elliptic curve. 163
Embedded prime divisor. 14
Etale morphism. 153
Euler-Poincare characteristic. 159
Exact sequence of /-inductive systems. 26
Exact sequence of presheaves. 28
Exact sequence of sheaves. 29
Exceptional curve. 205
Exceptional curve of the first kind. 169
Exceptional set. 205
Extension. 38
Factorization theorem of birational morphisms. 204
Family of transition functions. 120. 121
Fiber. 87
Fiber product. 84
Fibration. 184
Finite covering. 96
Finite covering space. 96
Finite morphism. 96
Finitely generated. 33
Finitely generated F-scheme. 89
Finitely generated extension. 3
Finitely generated morphism. 89
First theorem of Bertini. 182
Fixed component. 125
Fixed part. 125
Flasque sheaf. 47
Flat. 71. 88
Formula of Riemann-Hurwitz. 162
Free basis. 33
Function field. 90
Generalization. 81
Generic fiber. 184. 185
Generic point. 63
Genus. 160
Geometric genus. 160. 196
Geometric local ring. 184
Going-up theorem. 12
Graded module. 105
Graded ring. 105
Graph. 100
Ground field. 159
Height. 17
Higher direct image. 42
Hirzebruch surface. 218
Homogeneous. 105
Homogeneous coordinates. 110
Homogeneous ideal. 105
Homogeneous part. 106
Homogeneous spectrum. 106
Homomorphism of degree 0. 112
Homomorphism of presheaves. 27
Homotopic. 38
Hyperplane. 111
Hypersurface. 111
/-adic topology. 136
Ideal quotient. 14
Ideal sheaf. 33
Idealization. 79
Image. 180
Index. 122
Inductive limit. 25
Inductive system. 25
Infinitely near point. 194
Infinitely near point of the first order. 194
Injective Module. 37
Injective resolution. 38
Injective system. 37
Integral. 7
Integral closure. 9
Integral extension. 7
Integrally closed. 9
Integrally closed integral domain. 9
Intersection matrix. 188
Intersection number. 165. 167
Inverse image. 31. 32
Invertible sheaf. 115. 120
Irrational pencil of curves. 175
Irrational ruled surface. 216
Irreducbility theorem. 182
Irreducible. 62
Irreducible component. 63
Irreducible decomposition. 64
Irreducible divisor. 121
Irreducible ideal. 13
Irreducible pencil. 175
Irregularity. 197
Irrelevant ideal. 105
Jacobian criterion of nonsingularity. 142
^-module. 124
Kernel. 29
Kodaira dimension. 208
Krull dimension. 11. 62
Liiroth's theorem. 20
Lemma of Artin-Rees. 17
Length. 4. 62
Leray spectral sequences. 54
Lifting. 147
Lilnear subsystem. 124
Linear pencil. 182. 186
Linear system. 124
Linear topology. 133
Llinearly equivalent. 122
Local homomorphism. 24
Local intersection multiplicity. 167
Local ring. 11
Local ringed space. 31. 32

INDEX
245
Locally finitely generated morphism. 8
Locally free. 33
Locally free sheaf. 120
Locally Noetherian scheme. 97
Loop. 215
Lying-over theorem. 12
m-adic completion. 137
Maximal condition. 9
Maximal system of transcendental
elements. 4
Minimal algebraic surface. 207
Minimal model. 207
Minimal section. 219
Module. 32
Module of differential forms. 144
Monoidal transformation. 169
Morphism of affine algebraic sets. 77
Morphism of affine schemes. 62
Morphism of inductive systems. 25
Morphism of local ringed space. 32
Morphism of schemes. 81
Morphism of sheaves. 27
Multiple fiber. 186
Multiplicative group. 24
Multiplicity. 169. 186
Presheaf. 26
Presheaf (or a sheaf) of j/-modules. 27
Primary ideal. 13
Prime divisor. 14
Prime divisor decomposition. 14
Prime ideal sequence. 11
Problem of Liiroth type. 21
Problem of Zariski type. 21
Projective algebraic variety. 110
Projective bundle. 211
Projective line. 103
Projective Module. 37
Projective module. 37
Projective scheme. 110
Projective space. 110
Projective transformation. 112
Projective transformation group. 112
Proper. 92
Proper morphism. 93
Proper transform. 101. 170. 171. 205
Pull-back. 127
Purely transcendental extension. 3
Quasicoherent. 33
Quasicompact. 62
Quasicompact morphism. 89
N-genus. 196
«-section. 188
Neron-Severi group. 197
Nakai's criterion of ampleness. 169
Nakayama's lemma. 16
Noether's normalization theorem. 10
Noetherian domain
Noetherian. 10
Noetherian space, 62
Nonsingular. 140
Nonsingular algebraic variety. 141
Normal algebraic variety. 97
Normal ring. 15
Normal scheme. 97
Normal sheaf. 150
Normalization. 9. 98
Normalization morphism. 98
Nullstellensatz of Hilbert. 15
Numerically equivalent. 197
Numerically equivalent to zero. 197
Open immersion. 81
P'-bundle. 215
P1-fiber bundle. 215
P'-fibration. 215
Pencil. 175
Pencil of curves. 175
Picard group. 120
Picard number. 197
Polar divisor. 122
Polar part. 122
Ramification index. 153
Rational algebraic surface. 216
Rational curve. 163
Rational function. 74
Rational function field. 3. 74
Rational functions. 3
Rational mapping. 99
Rational mapping associated with linear system. 125
Rational point. 90
Rational surface. 216
Reduced. 74. 82
Reduced divisor, 121
Reduced form. 82
Reducible. 62
Reducible ideal. 13
Reduction. 87
Refinement. 45
Regular. 74
Regular extension. 6
Regular local ring. 139
Regular system of parameters. 139
Relatively minimal model. 208
Restriction morphism. 26
Riemann-Roch Theorem. 160. 195
Ringed space. 31
Ruled surface. 215
Scattered sheaf. 47
Scheme. 81
Second theorem of Bertini. 185
Section. 28. 188. 212
Semi-local ring. 184

246
INDEX
Separable extension. 5
Separated. 88. 134
Separating. 134
Separating transcendence basis. 5
Serre duality theorem. 155
Set of sections. 40
Sheaf. 27
Sheaf of differential forms. 145. 150
Sheafification. 28
Sheafifying space. 28
Shortest expression by primary ideals. 14
Simple normal crossings. 205
Singular fiber. 186
Singular fibers. 215
Singular locus. 141. 142
Singular point. 140
Smooth. 140
Smooth morphism. 152
Specialization. 81. 87, 91
Spectral sequence. 51
Spectrum. 61
Stalk. 27
Standard scattered resolution. 55
Stein factorization. 178
Structure morphism. 81
Structure sheaf, 66
Subscheme. 88
Support of a divisor. 125
Symmetric (¾-Algebra. 211
System of local coordinates. 152
System of parameters. 24. 139. 151
Tangent sheaf. 150
Tautological invertible sheaf. 211
The minimal model. 208
Theorem of Bezout. 167
Theorem of Tsen. 215
Total transform. 101. 170. 171
Transcendence basis. 5
Transcendence degree. 5
Transcendental. 3
Transition matrices. 211
Trivial. 219
Trivialize. 120
Universally closed morphism. 93
Universally normal ring. 184
Unrarmfied. 153
Unramified morphism. 153
Upper semicontinuous. 102
Valuation ring. 15
Variable part. 125
Very ample. 127
Virtual genus. 160
Weil divisor. 121
Zariski tangent space. 148
Zariski topology. 106
Zero divisor. 122
Zero part. 122