Текст
                    Lectures on
DEFORMATIONS OF SINGULARITIES
By
MICHAEL ARTIN
Notes by
C. S. Seshadri
Allen Tannenbaum
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
BOMBAY
1976


TATA INSTITUTE OF FUNDAMENTAL RESEARCH, 1976 No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 PRINTED IN INDIA BY AWL D. VED AT PRABHAT PRINTERS. BOMBAY 400 004 AND PUBLISHED BY THE TATA INSTITUTE OF FUNDAMENTAL RESEARCH.
CONTENTS Page Introduction i Part I: Formal Theory and Computations 1 1. Definition of deformations 1 2. Iarrobino's example of a O-dimensional scheme which is not a specialization of d distinct points 4 3. The meaning of flatness in terms of relations 7 4. Deformations of complete intersections 11 5. The case of Cohen-Macaulay varieties of codimension 2 (Hilbert, S chaps) 16 6. First-order deformations of arbitrary X: Schle s singer1 s T. ... 25 7. Versal deformations and Schles singer1 s theorem. 34 8. Existence of formally versal deformations. 40 9. The case that X is normal. 46 1Q. Deformation of a quotient by a finite group action 51 11. Deformation of cones. 52 12. Theorems of Pinkham and Schles singer on deformation of cones. 60 13. Pinkham1 s computation for the cone over a rational curve in IP . 67 Part II: Elkik1 s Theorems on Algebraization. 80 1. Solutions of systems of equations 80 2. Existence of solutions when A is t-adically complete 82 3. The case of a henselian pair (A,0C). 93 4. Tougeron1 s lemma. 100 5. Existence of algebraic deformations of isolated singularities. .... 104
Introduction These notes are based on a series of lectures given at the Tata Institute in January-February, 197 3. The lectures are centered about the work of M. Schlessinger and R. Elkik on infinitesimal deformations. In general, let X be a flat scheme over a local Artin ring R with residue field k. Then one may regard X as an infinitesimal deformation of the closed fiber Хл = X X/Dx Spec(k). Schlessinger1 s main result proven here in part 0 Spec(R) (for more information see his Harvard Ph.D. thesis) is the construction, under certain hypotheses, of a "versal deformation space" for X . He shows that -j a complete local k-algebra A = limA/m and a sequence of deformations X over Spec(A/m ) such that the formal A-prescheme 3Cs HmX is versal in this sense: For all Artin local rings R, every deformation S/Spec(R) of XQ may be obtained from some homomorphism A -• R by setting Note that by 'versal deiormation11 we do not mean that there is in fact a deformation of X_ over A. The versal deformation is given only as a formal scheme. However, Elkik has proven (cf. "Algebrisation du module formel d'une singularite isolee" Se'minaire, E. N. S. , 1971-72) that such a deformation of X over A does exist when XQ is affine and has isolated singularities. We give a proof of this result in these lectures. Finally, some of the work of M. Schaps, A. Iarrobino, and H. Pinkham is considered here. We prove Schaps1 s result that every Cohen- Ma caul а у affine scheme of pure codimension 2 is determinantal. Moreover, we outline the proof of her result that every unmixed Cohen-Macaulay scheme { of codimension 2 in an affine space of dimension < 6 has nonsingular deforma- deformations. For more details
see her Harvard Ph. D. thesis, "Deformations of Cohen-Macaulay schemes of codimension 2 and non-singular deformations of space curves". We also reproduce Iarrobino's counterexample (cf. "Reducibility of the families of 0- dimensional schemes on a variety", University of Texas, 1970) that not every 0-dimensional projective scheme (in IP for n > 2) has a smooth defor- deformation. Finally, we give some of Pinkhamfs results on deformations of cones over rational curves (cf. his Harvard Ph. D. thesis, "Deformations of algebraic varieties with <E action), m There is of course much more literature in this subject. Two papers relevant to these notes are Mumford's "Pathologies IV" (Amer, J. Math. , Vol. XCVII, p. 847-849) in which expanding onlarrobino's methods he proves that not every 1-dimensional scheme has nonsingular deformations, and M. Artin' s "Versal deformations and algebraic stacks" (Inventiones mathematicae, vol. 2 7, 1974), in which is shown that formal versality is an open condition. Thanks are due to the Institute for Advanced Study for providing excellent facilities in getting this manuscript typed. DEFORMATIONS OF SINGULARITIES part I. Formal Theory and Computations. $1. Definition of deformations. We work over an algebraically closed field k. Let X be an affine scheme, X«^* А.П. Let A be a finite (i.e., finite- dimensional/k) local algebra over k, so that A~ k[t^ . . . , t^] /0L with J&C = (t , . . . ,t ). Let T = Spec A. Definition 1.1. An infinitesimal deformation X (or X ) of X is a scheme flat/T together with a k-isomorphism X X^, Speck — X. More generally, suppose we are given a commutative diagram where R is a ring of finite type over к and X is a scheme with closed fiber isomorphic to X. We then say that X is a family of deformations or a deforma- deformation of X over R. Remark 1. 1/ X is necessarily affine. In fact J a closed immersion X ^а!1 ( = Spec A [X, . . . ,X ]) such that its base change with respecf to the morphism Speck-* T (representing the closed point of Spec A) is the immersion Let &= coordinate ring of X and <У= к [XJf . . . , Xj /I, with xi the II
canonical images of Xi in &. To prove the remark, it suffices to prove that if Xе-* A is an affine scheme (over A) imbedded in АП, A' is A A A' a finite local algebra/k such that 0-*J—A'—A — 0 is exact with J = 0 (J is an ideal of square 0) and X , is a scheme/A» such that XA' XSpecA' SPecA:rxA> then the immersion Xе-* A** can be lifted to an immersion XA,^-~ A*, . (This reduction is immediate since the maximal ideal is nilpotent. Say that m^t = 0, and take J = m^",1. . . ). We have an exact . sequence where x XA« denotes the structure sheaf of X . Now I2 = 0 since J2 = 0. A This implies that I, which is a priori a(sheaf of) & module(s), is in fact XA' a module over & /I, i. e. , it acquires a canonical structure of coherent A1 Crx -module. Since Хд is affine, it follows that н\х ,, I) = 0 (for it is A a = h\xa,I)). Hence 0-H°(XA,,I)-H°(XA,. X is exact, i.e., in particular H°(X GT )-* H°(X, cT )-0 is exact Let x XA' XA ' Ц be the coordinate functions on Хд which define Хдс-~ 1°, The x. can be lifted to |. € H (XA, , &x ). Then the g. define a,morphism ? :X , - АП . It A' A follows at first that ^ is a local immersion; for this it suffices to prove that' | generate the local ring & at every closed point x of X , . Let 1 A',x 1 be the stalk of the ideal sheaf I at a closed point x of X. We have x 0 — I -+ & % — &* -~ Q (&* , , &\ represent the local rings at x x A ,x A#x A ,x ' A, x of XAI and XA respectively). We have I =J- сУ . Now j- 8 =j* 8 .A A xa,x it. for j € J аш? 8., 8_ in & , such that their canonical images in & are 1 c, A f X A, X the same. Let S be the subalgebra of & t generated by g. over A1. A $ x l Then we see that I = JS. Since J(^A' it follows that I C S. Since S maps onto & t given X € & t , J s € S such that X-s € I . This implies that Ay X А у X X X « S. This proves S = & . We conclude then that ^ :X ,^1 , is a local A x A A v x X ,^1 , A A immersion. But ? is a proper injective map (since X е-* A» A A is a closed im- mersion). From this it follows that 4 is a closed immersion. This proves the Remark. Note that in the above proof we have not used the fact that XA, is flat/A' . Given the closed subscheme Xе-* Ж, let us define the following two functors on the category of finite local algebras over k. (Def. X) : (Finite local alg) - (Sets) II {Deformations of X /a}i- {isom. classes (over A) of schemes X flat/A and such that XA ® к- A (Emb. def. XH -{Finite local alg.) — (Sets) {Embedded deformations/A ь* {closed subschemes X of 1 flat/A, such that of X} . A A Xе—A by base change is the given X*~ A., }.
These should be called respectively infinitesimal deformations of X and infinitesimal embedded deformations of X. Then we have a canonical morphism of functors (Emb. def. X) - (Def. X). The above Remark says that f is formally smooth; that is, if A1 -*A -* 0 • is exact in (Finite local alg.), we have (Emb. def. X)(Af) - (Def. X)(A') I i (Emb. def. X)(Af) -* (Def. X)(A) (by definition of a morphism of functors), and the canonical map (Emb. def. X)(Af) - (Def. X)(A!) X (Emb. def. X)(A) \jjei. ада; is surjective. §2. Iarrobino's example of a O-dimensional scheme which is not a specialization of d distinct points. Given X as above, we can ask whether it can de "deformed" into a nonsingular scheme. Here by a deformation we do not mean an infinitesimal one, but a general family of deformations. Let us consider the simplest case of Krull dimension 0. Then X = Spec & where & is a k-algebra of finite dim- dimt2) ension d. If d = 1, e^k. If d = 2, O'srk X к or ©^*k[t]/(t ). If d = 3, we have (k3 or к X k[t]/t2 or k[t]/t3 or k[X,Y]/(X,YJ In our particular case the question is whether X can be deformed into d distinct points of A. . Now A. C-*JP~ and we see easily that this deformation implies a "deformation" of closed subschemes of IP , i. e. , if. X can be de- deformed into d "distinct points" we see (without much difficulty) that this can be done as an "embedded deformation" in A. and in fact as an embedded de- deformation in IP . Let Hilb denote the Hilbert scheme of 0-dimen- d sional subschemes ZeL-* IP such that if Z = Spec В then В is of dimdoverk. Then it is known that Hilb, is projective/k. Let U, denote the open sub- a a — scheme of Hilb corresponding to d distinct points, i. e. , those closed sub- subschemes of IP corresponding to points of Hilb , which are smooth. We see a that U is irreducible; in fact, it is the d-fold symmetric product of IP minus the "diagonals". Now if every O-dimensional subscheme can be deformed into a nonsingular one, then U, is dense in Hilb , and it would follow that Hilb , ad a is irreducible. We shall now give the counterexample (due tolarrobino) where Hilb d is not irreducible. It follows therefore that a O-dimensional scheme cannot in general be deformed to a smooth one. Theorem 2.1. Let Hilb, denote the Hilbert scheme of closed 0- - d,n dimensional subschemes of IP of length d. Then Hilb, is irreducible d, n for n < 2. For n > 2, Hilb. is not in general irreducible. ~ d, n Proof. We give only the counterexample for the case n > 2. Let СГ1 =k[x.,...,X ]/(X.,...,X )r+1. Let I be the ideal in (Х)Г/(Х)Г+1 (where u 1 nJ 1 n (X) = (Xy . . . ,XJ). Then I is a vector space over к and
Rank of i/k = Polynomial of degree (n-1) in r Rank of (У/к= " " " n in г. Now I is an ideal in &x and in particular an &% module. Moreover, if X с max ideal of &% then X-1 = 0. Hence &' operates on I through its residue field. In particular, it follows that any linear subspace (over k) of I is an ideal in 6Г' and hence defines a closed subscheme of Spec &x . Take now 9 = j Rank I or j Rank I-\ according as Rank I is even or odd and d = Rank(#"/V), where V is a subspace of I of rank 9. Hence 9= Polynomial of degree (n-1) in r and d = Polynomial of degree n in r. Let us now count the dimension of the set L of all linear sub spaces of I of rank = 9. 9 Then it is a Grassmannian and dimL = (j Rank I) or (j Rank I~j)(j Rank I+j) 9 according as Rank I is even or odd = (Polynomial of degn-1) in r = Polynomial of degree Bn-2) in r. Now if П is the subscheme of Hilb —- a d of "d distinct points" as above, then dimU, = d*n = n (Polynomial of degree n in r). Now if r » 0, we see that dimL > dimU,. 9 d Since L can be identified as a subscheme of Hilb ,, it follows now that U, 9 ad is not dense in Hilb,. To see that L_ is a subscheme of Hilb,,, note that a 9 a Spec &/1 as a point set consists only of one point. The family of subschemes of IP parametrized by L as a point is given by (x XL ). It suffices to 9 о 9 check that on (x X L ), we have a natural structure of a scheme Г such that 9 IP Г is a closed subscheme of IP XL and p" (x) is if the subscheme of Spec QT/l defined by the linear subspace of I corresponding to x. Then we see that p : Г — L is flat, for p is a finite morphism over L such that Ф" *s a sheaf of & -algebras; 9 1 LQ in particular, <Э" becomes a coherent sheaf over С . At every X€ L , 1 LQ 9 for &— ® <Э"Т , x/maxideal ( = &— ® k), the rank is the same. Since L_ Г Le Г 9 is reduced, this implies that & is locally free over & . In particular, Г is flat/L.. §3. Meaning of flatness in terms of relations. A module M over a ring A is said to be flat if the functor N^M ® is exact. <=> TorA(M,N) = 0 \/N/A. q <=> Tor^(M, N) = 0 V finitely generated N/A. Let us now consider the case when A is a finite local k-algebra. Then if N is an A-module of finite type, there is a composition series N = Nq JNX D. . . D^ = 0, such that N./N^l ^k. From this it follows immediately that M flat/A <=5> Tor (M,k) = 0 (if A finite local alg/k). Let X = Spec & , A finite local algebra and & an A-algebra of iinite type so that we have exact with P = A [x] (X = (X , . . . ,X )). Tensor this vrith k. Then we have
Let X = Spece;, X%X 0k, I ideal of X in A . Then к A. XA is flat/A <=> Tor.(e\ ,k) = 0, A 1 A <=> I ® к = I. A Take a presentation for the ideal I in P , i. e. , an exact seque A A IA (*) Then we have (because of the above facts): & }± A-flat <=> the above pre- 1 —— ———- ————— — ——— ——— ^ sentation for I tensored by k, gives a. presentation for I, _L je. , tensored by к gives again an exact sequence that its ®A к is the given exact sequence Pm — P -» & -* 0. Note that this need not imply that I ® к = I if I = Ker(PA - &A and I = Ker(P-c^). A A A A Suppose we are now given a "complete set of relations" (or a presen- presentation for I) between ?\s, i. e. , an exact sequence Then giving a^ lifting of these relations to that of f! ¦ s (or I ) is to give 1 A A A A A which extends the exact sequence РШ -* P - & - 0, which is a complex A A A at PA, i.e., ImPACKer(P^-PA) and such that (**) lifts (*). In this situation we have the following Proposition 3. 1. Suppose k /\ 0 0 (**) is exact and <**> is a complex such that the part Suppose we are given! <ЭГ= Kfxi/tf,,. . .,f ) and liftings (f!) of (f.) to elements in A.[x]. \ u J 1 m li m Let I = (f1f . . .,f ), IA = (f!,. . . , f ) and €ГД = A [x] / (?\ , . . . , f! ). These ? A "* A "* A "* 1 m A 1 m A x m .; \ data are equivalent to giving a lifting P™ - Рд - &A - 0 of the exact sequence V,, ^^ ^ (^} y = {^ Then & ^ ^^ -¦ CC -* 0, i. e. , to giving an exact sequence P -* P -* & -*• 0 such Proof. Suppose first that
10 и (**) is exact not merely a "complex at Pm". Then we claim that the flatness of A & over A follows easily. For then (**) can be split up as: Corollary Let &= k[x]/(f . , f ) and & = A [x]/(f' . . , f ) l m A 1 m where f! are liftings of f.. Then is A-flat <=> every relation among (f.,. . . , f ) lifts to a relation 1 m among exact. Therefore P* ® к - L. ® к-* 0 and L ® к - Pm ® к - I ® к -* 0 are exact. А А А л л This implies that 1Д 0 к = Coker(k ® P^ - к ® P™), i. e. , cokernel is pre- preserved by к ® . On the other hand, I = Coker(P — P ). Hence 1Д ® к = I. From this it follows that & is flat/A as remarked before. Now the hypotheses] of our proposition amount to the fact that all relations in I can be lifted to I . Given a relation in I , i.e., an m-tuple (X?, . . . , X' ) such that SX!f' =0, A -i m l i this descends to a relation in 1 by taking the canonical images of X^ in P. Take a complete set of relations for I , i. e. , an exact sequence A f' m U) (i1 need not be I), then from our above argument this descends to a complete set of relations for I, i. e. , tensoring (i) by к we get an exact sequence (ii) Pi? -*Pm-P-(^-0. Consequently, we have already shown in this situation that we must have ©*" to be A-flat. q. e. d. Remark 3. t. It is seen immediately that O: flat/A => I flat/A. A A For, the exact sequence 0 — I — P — €f -* 0 by tensoring by к gives A A A 0 0 0 II II II Tor (У -k)-Tor (I k)-Tor1(P.,k)-*Tor.(©'..kJ-l-P, -^-0. ?• ¦"> LA 1A 1A к This implies that Tor (I ,k) = 0, hence that I is flat/A, by a previous 1 A A • Remark. Repeating the procedure for <Э^А, we see by successive reasoning ' A that any resolution for & lifts to one for <У . A $4. Deformations of complete intersections. Let Xе- A. be a complete intersection. Let d = dimX (Krull dim) and I = (fx, . . . , fQ_d) the ideal of X. Let & = P/l, where P = P = к [x ,. . . , X ] Then it is well known that the "Koszul complex" gives a resolution for & , i. e. , the homomorphisms being interior multiplication by the vector The criterion for flatness can again be formulated as follows:
12 13 {iy . . .|»f ,) « P » (e. g. , P — P is the map (\ ,. . . , X .) *~ L.X.f.). The image of ЛрП" in Pn" gives the relations in I, which shows that the relations among the f. are (generated by) the obvious one-s, i. e. , f.z.-f.z. = 0. Let A = k[t]/(t ) (which we write A = k+kt with t = 0). Then deformations of X over A are called first order deformations. Let I be the ideal in /К Гх,,. . . , X 1 iA = where g. « P.. We claim that for arbitrary choice of g. « P., & =PA/IA 1 к -—-_ — 1 К А A A is flat over A (of course we have seen that any deformation X of X can be defined by I for suitable choice of g.). This is an immediate consequence of the fact that the above explicit relations between f. can obviously be lifted to relations between (f.+g.t). This proves the claim. Thus to classify embedded first order deformations it suffices to write down conditions on (g.)# (g!. ) in P so that in P the ideals ((f.+g.t)) and l 1 A 1 1 ((l+g[t)) are the same. We claim; ((f.+gjt)) = ((f.+g! t)) <=> g.-g« € I. To prove this we proceed as follows: (r X r) matrices (a..) and (p..) with coefficients in P such that , i.e.. (a..-Id)C ) = @), and (a) (о. .) J V f = Y f (b) (a..) + (P..) Since the coordinates of the relation vectors are in I it follows from (a) above that the elements of (a..-Id) are in I, i. e. , (a) => (a..) = (Id)(mod I). Then (b) implies that (mod I). Hence ((f.+g!t)) С ((f.+g.t) => (g.-g!) € I. Hence ((f.+g.t)) = ((f.+g!t)) => (g.-g^c I. Conversely, suppose that (g.-g!)c I. Then there is a matrix (p..) such that (P..) • • f _ г = . gi g tf.+g! t)) С ШЛЯЛ)) <=> (Set r = n-d.) Hence, (Id) V \ gr + (pjj) Y \ gi" _g'r_ г = (Z Q;; Г Г tB a..g. + 2 Taking (а_) = Id we see that the conditions (a), (b) are satisfied, which implies that ((f.+g.t)) = ((Г+g! t)). This proves the claim and thus we have classified
14 15 all embedded (in A. ) first order deformations of X. Now to classify first order deformations of X we have only to write down the condition when two embedded deformations X , X1 are isomorphic A A over A. Let в be an isomorphism X. 'X X\ . By assumption, X в к = X' в к = X, i. e. » their fibres over the closed point of Spec A are We denote {of course) by X the canonical coordinate functions of X «-» A. = Soec A [X ,..., X J. Let X^ = 6*(Xy). Then we have x;=xv+?v(x)t for some polynomials <P (X). Hence to identify two embedded deformations of X we have to consider the identification by the above change'of coordinates. Then By Taylor expansion up to the'first order, we get = f (X) + t {g.(x) {g.( v=l Hence (f.+g.t) and (f.+g! t) define the same deformation of the first order up to change of coordinates above, which is equivalent to the fact that there exists (CD ,,. . , <pn) such that V У эх, эх 1 n 9f 9f г ... г ЭХ, ЭХ _ 1 n "9 Г (*) Recall that &- P/l and X = Spec &. Then embedded first order deformations are classified by Ф =©'Э...9<У (n-d times), which is an &- module . To classify all deformations, consider the homomorphism of P modules T Jac. , di. 3f. 3f. & whose matrix is fef-)j "ггг* the images of 3V in P/I. dX. dX. dX. Let us call the quotient & " /imC by this map T. This is an С-module and we see that its support is located at the singular points of X. In particular, if X has isolated singularities, T is a finite dimensional vector space/k. For example, consider the case that X is of codimension one, i. e. , defined by one equation f. Then '¦"МЦ э5Г>- 1 n The cone in 3-space has equation f = Z -XY, and if char к f 2, T = к [X, Y, Z] / (f, - Y. -X, 2Z) 3- k. Thus a universal first order deformation is given by Z2-XY+t = 0.
16 17 $5- The case of Cohen-Macaulay varieties of codim 2 in A (Hilbert, Schaps). The theorem that we shall prove now was essentially found by Hilbert. This has been studied recently by Mary Schaps. Let P be as usual the polynomial ring P = k[X ,. . . , X ]. Let (g..) be an (r X r-1) matrix over P elr-l r,r-l Then F^ ...,6r) . Let 6. = (-1I det((r-l) X (r-1) minor with ith row deleted). r,l'** gr,r-l = 0; for, det T,l'- gr,r-l' gr,l = 0, etc. is exact, i. e. , it gives a resolution for P/J. 2) Conversely, suppose given a Cohen-Macaulay closed subscheme Xе-* АП of codim 2. Let X = V(J); then P/J has a resolution of length 3 r-l(gii) rV'-'V which will be of the form 0 - P —±*+ P * P -* P/ J -* 0, because hdpP/J = 2. (Depth P/j+hdpP/j = n, depth P/J .= n-2, => hgP/J = 2. ) Note that f. need not be 6. as defined above. Then we claim that we have an iso- isomorphism О 0 -* r'6l'---'6r> P ist. This implies that the sequence V Fi V (g..) has rank (r-1), or equivalently, J some x € A such that (g..(x )) is of rank (r-1). Theorem 5. 1 (Hilbert, Schaps). 1) Let (g..) be an r X (r-1) matrix over P and 6. its minors as defined above. Let J be the ideal F^ . . . , б ). Assume that V(J) = VF.,. . . , б ) is of codim > 2 in A , Then X = V(J) is Cohen-Macaulay, precisely of codim 2 in A . Further, the sequence r Fl ,6 ) i. e. , J a unit u € P such that f. = иб.. 3) The map of functors (Deformations of (g..)) — (Def. X) is smooth, i.e., (i) deforming (g..) gives a deformation of X, (ii) any deformation of X can be obtained by deforming g.., and (iii) given a deformation X of X ij A defined by (g..) over A[x], A1 -*A-*0 exact and a deformation X , of X inducing XA, 4 (g1..) over A1 [xl which defines Хд| and the canonical A —' ij L J A1 image of (g|L) in A is (g ). that Proof. A) Let (g..), 6., etc., be as in A). " We shall first prove o-p-Vp. is exact, assuming codimX> 2. - This will complete the proof of A). For, it follows hd P/J<2. On the other hand, since codimX>2, depth P/j<(n-2).
18 19 But depth P/J+hd P/J = n. This implies that. dimP/j = depthP/J = (n-2) and hd P/J = 2, which shows that X. is Cohen-Macaulay of codim 2 in A . Since V(J) к A , the 6.'s are not all identically 0, hence 0 —P " -* P is exact. Further we note that at any x / V(J), (g. .(x )) is о ij о of rank (r-1) and in fact one of (б.,. . . , 6 ) is nonzero at x and hence a 1 г о unit locally at x . This implies that РГ" — рГ -* P is split exact locally at —___ Q _________ x (i. e. , if В is the local ring of A. at x , then tensoring by В gives a о о split exact sequence). Because of our hypothesis that codim V(J) > 2, if x is a point of Ж represented by a prime ideal of height one and В its local ring, then tensoring by В makes P — P — P exact (i. e. , the sequence is exact in codim 1). Let К = Кег(РГ-Р). We have KQ P* such that 0 — P " — K, and, 0 - K— P — P exact. Tensoring by В as above, it follows that P 0 В —К ® В is an isomorphism (tensoring by В is a localization); i.e. , the inclusion P С К is in fact an isomorphism in codim 1. Since рГ~ is fre$ sections of P defined in codimension 1 extend to global sections. Moreover, К is torsion-free. Therefore, the fact that P С К is an isomorphism in codimension 1 implies that it is an isomorphism everywhere. Therefore, P " -* P ¦* P is exact, and this completes the proof of A). B) Let I = (f ,. . . ,fy) be a Cohen-Macaulay codim 2 ideal in P. Since hd P/l = 2, there is a resolution (*) "'•••'V (Here we should take the warning about using free resolutions instead of using projec- tive resolutions. ) Let the complex (*#) be defined by (**) r 0 -» РГ -^P-P/J-O, 6. being as before. The sequence (*) is split exact at every point x / V(I). This implies that some 6. is a unit at x, and hence by direct calculation that (**) is also split exact, and P/J = 0, at x. This means V(J)CV(I), hence codim V(J) > 2. Hence by A) it follows that (**) is exact. We shall now show that a unit u such that Take the dual of (*) and (**) (dual: Homp(M, P) = M ), i. e. , (*) (**)* <pV ) F.) We claim that these sequences are exact. The required assertion about the * {?)Л * г existence of и is an immediate consequence of this. For then P * (P ) 3 (S.) # r % r and P * (P ) are injective maps into the same submodule of (P ) of rank 1 which implies that (f) and F) differ by a unit. We note that {1±? l ff! P and -> Р 5r. P are split exact in codimension 1. Consequently, it follows from this and the fact that Horn (P/I,P) = Horn (P/J, P) = 0 that we have sequences
20 21 О -* р Conversely, let X be an infinitesimal deformation of X = VF ,. . . , 6 ). Lift the generators for I to I say (f , . . . ,f ). Then as we remarked before A 1 Г the exact sequence (*) can be lifted to an exact sequence г* and we must prove exactness at the P module. But we have that bnf. CKer(g..) and ImF.) CKer(g..) with equality at the localization of each prime ideal of height 1. Consequently, bnf. = Ker(g..) , ImE.) = Ker(g..) (same argument as was used above). This completes the proof of B). C) Let A be an Artin local ring over k, and P = A[X , . . . , X ]. Let (gl.) be an r X (r-1) matrix over P and (g..) the matrix over P such ij A lj that g!.*-*g.. by the canonical homomorphism A [x] -»k[xj. Define 5| analo- analogous to 6.. Suppose that codimVF! ) in 1 is of codim >^ 2 <=> codimVF.) 1 1 A l in A is of codim ]> 2 <=> codimVF.) = 2 because of A). Consider (**) 0 - г (б!) ? —-— Р — Р /I — О А А А А (**)¦ r-l^lj» r(fi> р J— p — р — р /I А А А А' А [Note that f. need not be (a priori) the determinants of minors of (gl.). J Let 6'. be the minors of (g!.), J the ideal F', . . . , 5' ). Then as we saw above l ij A 1 г (**) is exact and PA/JA is A-flat. Taking the duals of (**) and ———— A. A )' as before, it follows that there is a unit u in P . such that f. = u6! . This shows that A li I = J . Thus any deformation is obtained by a diagram of type (**). The A A assertion of smoothness also follows from this argument. This completes the proof of the theorem. E.) (*) 0-] •P-*P/I-*0. Now (#*) is lifting of (#). Of course (*) is exact and P* — P -*PA/JA"*° A A A. A is exact. Besides, (**) is a complex. This implies by Proposition 3.1 that PA~ -* PA — PA -* Рл /JA — 0 is exact and PA/IA is A-flat and by Remark 3. 1 A A A A A A A 7 (**) is exact. (The exactness of (**) can also be proved by a direct argument generalizing A).) This shows that any (infinitesimal) deformation (g!.) of (g..) as in A) gives a (flat) deformation X of X = SpecP/l and that X is "pre- A A sented" in the same way as X. Remark 5.1. Any Cohen-Macaulay Xе- of codim 2 is defined by the minors of an r X (r-1) matrix (g..) (this is local, note the warning about using free modules instead of projective ones). The g..1 s define a morphism пФ Mr(r-1) Let Q=k[x..], l<i<r, i< j< r-1, so that АГ^Г"^ = Spec к [Х..]. Then Ф (X..)=g... Consider (X..) as an r X (r-1) matrix over Q, and let Д. = (-1I det(minor of X.. with i* row deleted). Then the variety
22 23 V = V(A , . . . , Д ) -* A is called the generic determinantal variety defined by г X (r-1) matrices. It is Cohen-Macaulay and of codim 2 in A Then ф" (x) = X. This means that any Cohen-Macaulay codim 2 subscheme is obtained as the inverse image by A -»1 of the generic determinantal variety V of type r X (r-1). <f € Hom(A , A. " ) and (V € X , then im Gf> is a k-dimensional space. Hence Ф is determined by an arbitrary k-dimensional subspace of A. { = kerfl?) an (r-k)-dimensional subspace of A { = Im CD) and an arbitrary linear map of an (r-k)-dimensional linear space onto an (r-k)-dimensional linear space. Hence Remark 5. 2. Other simple examples of codim 2, Cohen-Macaulay dimX = = (r-k)k+(r-k)(k-l)+(r-kJ. X are (a) any 0-dimensional subscheme in A , (b) any 1-dimensional reduced X in A , and (c) any normal 2-dimensional X in A . More about the generic determinantal variety. Now let denote the determinantal variety У(Д.) = V defined above. Then any infinitesimal deformation X of X is obtained by the minors of a matrix (X. .+m..) where m.. € mA Гх. Л, where mA is the maximal ideal of A. Set X|. = X. .+m... We see that X.. »-» X!. is just change of coordinates 4 4 4 lJ 4 in A^, i.e., A[X..] =A[X!.]. This implies that X is rigid, i.e., every de- formation of X is trivial. The singularity of X: X can be identified with the subset of Ж = Horn.. (A , A ) of linear maps of rank < r-2. Now linear r - G = GL(r) X GL(r-l) operates on A ~ in a natural manner. Let X к. denote the subset of A of linear maps of rank equal to (r-k). We note that X is an orbit under G and hence is a smooth, irreducible locally closed subset of A To compute its dimension we proceed as follows: If Suppose к = 2, i. e. , consider linear maps of rank (r-2). Then X is obviously open in X and dimX2 = (г-2J+(г-2)+(г-вГ = (r-2J+l+(r-s) = (r- It follows that codim X = 2. It is clear that X is dense in X (easily seen that every x € X is a ? , . о specialization of some x tf. X ). This implies that X is irreducible. Further- Furthermore, X is smooth, and in particular reduced. By the unmixedness theorem, since X is Cohen-Macaulay, it follows that X is reduced. Hence X is a subvariety of АГ*Г~1}. Let X1- = X UX ... be the s.$t,p|?all linear maps of rank ? (r-3). Clearly X' is a closed suoset of X. To compute its dimension it suffices to compute dimX-, as we see easily that X. is a dense open set in X' Now dimX3 = (r-3K+(r-3J+(r-3J = (r-3)(r+2). Hence codimX^ (in АГ*Г~ ') = 6, for r > 3. (If r = 2 it is a complete inter!-* section; X is the intersection of two linear spaces and hence X is smooth. )
24 25 Hence codirnX. in X is 4. We claim also that every point of X' is ———— j — — j singular on X. To prove this, suppose for example 0 € X . Since G acts transitively on X-, we can assume it is the point 9 = 1 0 0 0 10 0 0 1 ЭД. Then = 0 Vk, 1. Thus 9 € X is smooth <=> 9 € X . Thus the generic determinantal variety has an isolated singularity if and only if r = 3, in which case dimX = 4, XC A6 ( = A**1"*). We thus get an example of a rigid isolated singularity (a 4-dimensional isolated Cohen- Macaulay singularity). Proposition 5.1 (Schaps). Let X c-* A (d _< 5) be of codim 2 and Cohen-Macaulay. Then X. can be deformed into a smooth variety. Proof. We give only an outline. Since X. is determinantal, it 0 is obtained as the inverse image of X in A by some map Now codimX in A -¦ A is 6. "Perturbing" (g..) to (g!.)=<p!, the map , defined by (g!.) can be made "transversal" to X. This implies that qp1 (A )PlSingX = 0 [ Sing X is of codim 6] and in fact that q> ' ~\x) is smooth. Remark 5. 2. It has been proved by Svanes that if X is the generic determinantal variety defined by determinants of (r X r) minors of an (m X n) matrix, then X is rigid, except for the case m = n = r. $6.. First order deformations of arbitrary X and Schlessinger1 s T . Let X = V(I) and A=k[t]/(t2). Let I = (f ,. . . , f ). Fix a presenta- presentation for I, i. e. , an exact sequence (*) Let I = (f' ..,f ) where f| = (f.+tg.), g. € P. Then X. = V(I) is A-flat Aim iiii AA if (*) lifts to an exact sequence (**) , (r!.) (f!) A A A A A In fact we have seen (cf. Proposition 3.1) that in order that (**) be exact it suffices that (**) be a complex at Pm, i. e., _ A X = V(I ) is A-flat iff there is a matrix (r!.) over P extending (г„), such that Set r!. = r..+ts.., where s..€ P. Then X is A-flat iff J (s..) over P such that the matrix product (f+gt)(r+st) = fr+t(gr+fs) = 0,
27 where f = (f.),. . . , г = (г..). Now fr = 0 since (*) is exact. Thus the flatness of X over A is equivalent with the existence of a matrix s = (s..) A ij over P such that <*) (gr)+(fs) = 0. j : (First order def. of X) -* H°(X, N__) is injective, which implies, since j is surjective, that j is indeed bijective. The proof that I = J is immediate; from the computation in §4 we see that A A Consider the homomorphism <=> J (m X m) matrices (a..) and (p..) over P such that <g):Pm-*P defined by (g..). Then condition (*) implies that (g) maps Imp under the homomorphism (г..) : P -¦ P into the ideal I. Hence (g) induces a homo- homomorphism (g) : РШ/1ш(г) = I - P/I, i. e. , g is an element of Homp(I, P/I) - Homp/l<I/l2, P/l) = Hom^ (I/I2, &x) = X the dual of the coherent Cv module I/I on X. The sheaf Horn— (I/I ,& ) = X <3"х X Nv is called the normal sheaf to X in A . Thus g is a global section of N. Conversely, given a homomorphism g : Horn (I, P/I) and a lifting of g to g : P -* P, then (g) satisfies (*). This shows that we have a surjective map of the set of first order deformations onto H (X, N ). Suppose we are given X two liftings (f+tg ) and (f+g_t) such that g , g define the same homomorphisms 1 ?* 12 of I/I into P/I, i.e., ~g ="g . We claim that if I = ((f.+tg1 .) and ic, A i 1, i J = ((f.+tg .)), then I = J , i. e. , the two liftings define the same sub- A 1 ut 1 A A. scheme of A . This will prove that the canonical map A 5l,m As was done in §4, if (g^ = (g2), i. e gli"g2i ~ Then there exists (p..) such that (a) and (b) are satisfied with (a..) = Id. This implies that J QlA* In a similar manner I Q J , which proves that A A. A A J = I . Thus we have proved Theorem 6.1. The set of first order embedded deformations of X in Ж is canonic ally in one-one correspondence with N (or H (NY))--the X X normal bundle to X of ([ Remark 6.1. Suppose now we are given 0 - eA1 -*Af -*A-0 exact
2В where A1 , A are finite Artin local rings such that rk, (cA1 ) = 1 (in particular it follows easily that t is an element of square 0). We see easily that eA1 has a natural structure of an A-module and in fact eA1 ~A/m as A-module. """" A (Given any surjective homomorphism A1 -* A of Artin local rings by successive steps this can be reduced to this situation. ) Suppose now that X. is a lifting of X defined by V(I ), I = (f' . . . , f' ) with f! s as liftings of f., X = V(I), A A l m i l I = (fj,. . . , f^). We observe that even if A is riot of the form k[t]/t , in order that X be flat/A andbe a lifting of X it is neccessary and sufficient that A there exists a matrix (r!.) such that l' * '' ' fm)(rij) = ° (fl are liftings of f ). Suppose now we are given two liftings X^, and X|, of Хд, flat over A defined by V(I^,) and V(l|,): Now XA is defined by an exact sequence (*) such that 0 к gives the presentation for X. An easy extension of the argu- argument given before (cf. Proposition 3. 1) for characterization of flatness by lifting of generators for I shows that X' and X are flat/A. They are presented respectively by (a) exact 29 p A' such that (a 0 , A) and (p 0 A) coincide with (*) and in fact it suffices A A that (g)(s) and (g1 )(s' ) are zero and s, s1 are liftings of r. We can write g! = g.+eh., with h. с Р, and g! = g.., s!. = s..+et.., with t.. с P. ij iJ ij ij Then (g1 )(s' ) = 0 iff (g.)(t..) + (h.)(s..) = 0, and by the remark about eA1 as an A-module (or A' -module) it follows that the right hand side above is 0 if and only if (t) .)(..) О(г..) 0, i.e., j (t..) and (h.) over P such that this holds. Conversely, we see that given XAt flat/A1 and (-{-), we can construct X , flat/A1 by defining A A g! = g.+eh. and lifting of relations by s!. = s..+et... Now ("(¦) gives rise to N as before. Thus fixing an X1 flat/A1 which is a lifting of X flat/A л А А the set of flat liftings X , of X over A are in one-one correspondence with A A Nv (or H°(N_J) or in other words the set of flat liftings X , over X. is X X A A either empty or is a principal homogeneous space under N (ojr H (N )). In the case A = k, A! =k[t]/(t ) (or more generally A1 = A [~e~)) we us e X. , = X 0, A1 (resp. Хд| =ХА 0Д А1 ) as the canonical base point, and using А1 к ь A? A A ^ a this base point the set of first order deformations gets identified with Nv or H°(NX). Remark 6. 2. Compare the previous proof (cf. 5 4) for the calculation
ai зо of first order (embedded) deformations of X--a complete intersection in A . In that proof we required the knowledge of the relations between f.1 s where X = V(I), I = (f., . . . ,f ) whereas this is not required in the present proof (we 1 m assume X arbitrary). This is because in the proof for the complete inter- intersection we obtained a little more, namely, (First order embedded deformations of X) *¦-* {g.}, g. « CVr* The previous proof gives also the fact that N i^ i i X X free over <5^ of rank = codimX. Hence a better proof for the case of the X complete intersection would be to prove first the general case and then show that i/l is free of over & of rank = codimX (in A ) (this implies that N = X X Hom^ (I/I2, C_J is free of rank = codimX). Remark 6. 3. The above proof also generalizes to the computation for first order deformations of the Hilbert scheme or more generally the Quotient scheme (in the sense of Grothendieck) as well as the consideration of Remark 6.1 above. Schlessinger' s T : We have computed above the first order embedded deformations of X. Now to compute the first order deformations of X (as we did in §4) we have to identify two first order embedded deformations X , X , (Cl") A = k[t]/(t2) A A A when there is an isomorphism 9 : X ~ X1 which induces the identity auto- A A morphism X — X. As we saw before, the isomorphism 9 is induced by a change of coordinates in A , i. e. , if X are the canonical coordinates of A v A we started with and 9 (X ) = Xf then we have If a first order embedded deformation of X = V(I), I = (f.) vis given by A+g.t), then = fi(X)+t{S-^(|)v(X)} + tg.(X) i n df. = f.(x) + t{g.(x) + s -a5T< N and X1 the canonical image of are arbitrary elements of the n df. Let X be the canonical image of (g.) in N i л. n Ы. 'l*^»™ ^ NX* N°W ?1" •- coordinate ring of A , and the canonical image of Б - v=l precisely the image under the canonical homomorphism where в n represents the tangent bundle of A and |x denotes its restriction to X. We have a natural exact sequence ¦nx-° where О denotes the Kahler differentials of order one on a scheme X/k. The dual of this exact sequence gives an exact sequence =X
32 33 where в n -¦ N__ is the homomorphism defined above. We define T Ж X X X to be the cokernel of this homomorphism, so that we have -* N -*• T -*• О X ХХ * Thus we have proved Theorem 6. 2. The first order deformations of X are in one-one correspondence with T . X Remark 6. 4. Suppose that A1 = A[e] (i. e. , A1 = А в ек). Then the argument which is a combination of that of the theorem and Remark 6.1 above shows that Def(A' ) = Def(A) X Def(k[c]), i. e., the set of deformations of X over A1 which extend a given deformation over A are in one-one correspondence with the first order deformations of X. For the proof of this, we remark that a similar observation has been proved above for embedded deformations. Now identifying two embedded deformations XA, and X* which reduce to the same embedded deformation X4 over A, A1 A1 A the argument is the same as in the discussion preceding Theorem 6. 2, and the above remark then follows. Remark 6. 5. Deformations over A1 which extend a given deformation over A form a principal homogeneous space under Def(k[e]), or else form the empty set. Remark 6. 6. Note that if X is smooth, then T = 0. This implies that any two embedded deformations' X , X1 (A=k[e])c>f X are isomorphic over A (in fact obtainable by a change of coordinates in A.). Rema rk 6. 7. If X has isolated singularities, T as a vector space over к has finite dimension. For, it is a finite С -module with a finite set as support. Proposition 6.1. Suppose that X = X . Then tL - ExtL, (SI* X — "._ л л. Proof. Let X = V(I). Then the exact sequence i/i2 - can be split into exact sequences as follows: (i) (H) 0-c -I/I-F-0 _ _ .1 Since X = X ,, the set of smooth points of X is dense open in X, so that e, being concentrated at the nonsmooth points, is a torsion sheaf, in particular, Hom-ь, {t,&') = Q, Writing the exact sequence Hom(»,(9^Yl for (i), we get that crx x x 0 -Horn., (F, ©'„) - Hom(l/I2, &) - 0 is exact. ^x X || X N X
34 35 Writing the exact sequence Hom(« , С ) for (ii), we get that - F* - , &x) - 0 is exact (since Ext {Q\ n I , x Ix being free). Now T = coker(9 n I -*N__), and coker(9 n -* F ) = A A. I А А. Ix x above F ^-N__, and so A v t (Q A A . But as §7. Versal deformations and Schlessinger* s theorem. Let R be a complete local k-algebra with к as residue field (k-alg. closed as before). We write R = П where W. =Ъ1 is the maximal К _ where W. К К ideal of R. We are given a closed sub scheme X of А.П (note that definitions similar to the following could be given for more general X). Definition 7.1. A formal deformation X of X is: (i) a sequence ¦ xv. {X }, X = X where X • is a deformation of X over R , and (ii) iso- n n R JR. n n n morphisms X ®_ R , ~~ X , for each n. * n R n-1 — n-1 n Suppose that A is a finite-dimensional local k-algebra. Then a k-algebra homomorphism Cp : R-*A is equivalent to giving a compatible sequence of homomorphisms q? : R -* A for n sufficiently large. This is so because a (local) homomorphism Cp : R -» S of two complete local rings R, S is equivalent to a sequence of compatible homomorphisms <P : R/7K* -S/7fc!!, and in our case %,* = 0 for n » 0. Given a formal • n xv. о А deformation X_ of X and a homomorphism Q»:R-*A;X ®_ A (via К 'ПК n Q : R — A as above) is up to isomorphism the same for n » 0. It is a • n n deformation of X o by Spec A-* Spec R). • n n deformation of X over A. We define this to be X ® A (base change of X "~~~~ XV. XV. Definition 7.2. A formal deformation X of X is said to be versal if the following conditions hold: Given a deformation X of X over a finite A dimensional local k-algebra A, there exists a homomorphism Qf : R -* A and an isomorphism X ® A ~~X ; in fact, we demand a stronger condition as xv. A , Д follows: Given a surjective homomorphism A1 -* A of local k-algebras, a deformation X , over A1 a homomorphism 0?:R-»A and an isomorphism X О A ~ X , ® A, there is a homomorphism О' :R-*A' such that OP1 A. R "*~ A. ' lifts Cp and X 0 A1 is isomorphic to X , . (Note that it suffices to assume * xv. xv. A the lifting property in the case ker 0 is of rank 1 over k. ) Let F : (Fin. loc. k-alg. ) — (Sets) be the functor defining deformations of X, i. e. , F(A) = (isomorphism classes/A of deformations of X/A). Given a formal deformation X of X, let R G : (Fin. loc. k-alg. ) -* (Sets) be defined by G(A) = Horn, (R, A). Then we have a morphism j : G -¦> F K-alg. of functors defined by That a. formal deformation i? versal i_s equivalent to saying that the functor
36 j is formally smooth. Theorem 7.1 (Schlessinger). Let F : (Finite, local, k-alg. ) ~+ (Sets) be a (covariant) functor. Then there is a formally smooth functor (as above), i.e., Horn, (R,-)-F(-), it—axg. where R is a complete local k-algebra with residue field k, if A) F(k) = a single point. B) Given (е)-А1 -*A-»0 with A1 , A finite local к-algebras and (e) = Ker(Al -*- A) of rank 1 over к and a homomorphism (V : В — A let B1 = A1 X В = {(a, P) € A1 X В such that their canonical images in A A> are equal}. (SpecB* is the "gluing" of Spec В and Spec A' along Spec A by the morphisms SpecA-*SpecB and Spec A -* Spec A1. If Ф is surjective, i. e., Spec A -*• Spec В is also a closed immersion, this is a true gluing. ) Then we demand that the canonical homomorphism (*) is surjective. (Note that B1 is also a finite-dimensional local k-algebra. ) C) In B) above, take the particular case A = к and A1 = k[fi] (ring of dual numbers), then the canonical map defined as in (*) above F(B') -F(k[?]) X F(B) II II (> (F(B0])-F(k(>]) X F(B)) is bijective, not merely surjective. 37 D) F(k[c]) is a finite-dimensional vector space over k. Remark 7.1. One first notes that F(k[*]) has a natural structure of a vector space over к without assuming the axiom D) above: Given с с к, we have an isomorphism k[c] -*k[c] defined by e — с* г {of course 1-*1) which defines a bijective map By this we define "multiplication by с с к" on F(k[cJ). By the third axiom we have a bijection F(k[e x, с J) ~ F(k[e J ) X^ F(k[c J ) e (k[c.,t?] being the 4-dimensional k-algebra with с = e = 0 and basis 1, с с , e1?7)« We have a canonical homomorphism kfe ,?_]-¦ к [с] defined by €-¦?, €2"*e which gives a map F(k [c , ? ]) — F(k [z] ), so that we get a can- canonical map F(k[?]) XptF(k[?j) Here we use the fact that F(k[c] ek This follows by axiom C), and We define addition in F(k[e]) by this map, and then we check that this map is bilinear. This gives a natural structure of a k-vector space on
38 39 if axioms A), C) hold. Remark 7. 2. The versal R can be constructed with Ъ1 /Ж ~ F(k[-]). Then with this condition R is uniquely determined up to isomorphism. (Note: We do not claim that given X the homomorphism R-*A is unique. ) The A functor represented by R on (finite local k-alg. ) is called the hull of F which is therefore uniquely determined up to automorphism. Proof of Schlessinger' s Theorem. Let <? denote the full subcategory _________ _______________ т jj of (fin. loc. k-alg. ) consisting of the rings A such that Ъь = О. This category A is closed under fibered products. We will show the existence of R by finding j D J Dm n-1 n n-i a versal R for FJ? , for all n inductively. Take the case n = 1. <? is just the category of rings of the form A = к © V where V is a finite-dimensional vector space. So, (Jf is equivalent to the category of finite-dimensional vector, spaces. Let V be the dual space to F(k[e]), and put R = к © V. Then a map R_-*k[c] is given by a map V-»ck, i.e., an element of F(k[e]). So, Hom(R , k[_] ) = F(k [e] ). Since, by axiom C), F is compatible with products of vector spaces, R represents Suppose now that R is given, versal for F/C ., and let u , € n-l n-1 n-1 F(R .) be the versal element. Let P be a power series ring mapping onto R n-1 Choose an ideal J , n 0-*J -*p-*R n-1 n-1 which is minimal with respect to the property that u ., lifts to u^ € F(P/Jr and put R = P/J . We test (R ,u ) for versality. Let A1 -*A be a r n n n n surjection with length 1, kernel in Cn» and let a test situation be given: A1 I > A , u а1 с F(A! ) Form R1 = R X A' : П A Since P is smooth, a dotted arrow exists. By axiom B), there is a u1 € F(R' ) mapping to u and a1 . Since J was minimal, R1 cannot be a quotient of P. Hence im(P -> R1 ) % im(P -» R ) = R , and so R' -* R splits: n n n Let v1 be the element of F(R' ) induced from u by the splitting. The versality n will be checked if u1 = v1 , for then the map R^ -* R' —A1 is the required one. We can still change the splitting, and the permissible changes are by elements of Hom(Rn,k[>]) = Нот^,к[-]) = F(k[c]). By axiom C), F(R' ) = F(R ) X F(k[c]). Both u1 and v! have the same image un in F(Rn>. So, we can make the required adjustment. This completes the proof of Schlessinger' s theorem.
40 41 $8. Existence of formally versal deformations. Theorem 8.1. Let X(^A be an affine scheme over к with isolated singularities. Then for the functor F = Def. X =Def the conditions of Schles- singer1 s theorem are satisfied. In particular X admits a versal deformation. Proof. Since X has isolated singularities, we have ranK (Def. X) [k [e]] = ranlc (Т„) < oo. Hence it remains only to check the axioms B) and C). Axiom B): driven (e ) •* A1 -¦ A, q>x : В -* A a deformation X of X and two T A deformations X , , over Хд ^Al V" Let В1 = А1 X В. We need to find a deformation X . over B1 inducing А В the given deformations X , and X over A1 and В respectively. As usual we write (У__ = &. ,,..., etc. We now set & - & X_, ©1. We have X | А В А СГ В canonical homomorphisms B1 -*• В and B1 -• A1 . It is easily seen that ^ni ®T>t в % &o and ^>i ®tji A' % ^Ii • The оп1У serious point to check в а в а а а is that ©, is flat/B1 . Since B1 = A1 XA B, we have a diagram J3 A о-*(о-в' -B-o 0-(?)-A' -A-0 with exact rows, (c) is of rank 1 over k, so (e) «** к as B' modules. Similarly, II I It follows easily that the exact sequence 0 -* с & -* &' -*> & '-* 0 is obtained by tensoring 0-*(e)-*B! -*B-*0 with & , , and that с • & s% & . Now the flatness of Cf over B1 is a consequence of В Lemma 8.1. Suppose that 0 -* (e ) — В1 -* В -> 0 is exact with В' , В finite local к-algebras and rK(^) =1 (so that (e)^k as B1 module). Suppose that X_, = Spec O" is a scheme over B1 such that &. ®та1 В = <^та is В В В В В flat over В, with X = Spec <У a deformation of X = Spec & , and that В В к кег(Ф, -* & ) is isomorphic to & (as c7 module). Then & , is B1 В В к В В flat; in particular, X , = Spec Ol, is a deformation of X over B1 . В В * Conversely, if Ф^,, is B1 flat giving a deformation of X = Spec &* , В к ^Rt has an exact sequence representation as in the lemma . Proof. We have an exact sequence with e ОС ~ &. . We claim that this is obtained by tensoring (*) 0 -* (e)-^B' -* В -*0 by & t and in fact that it remains exact because of our hypothesis that is 5; ©: as B1 -module. For, tensoring (*) by &' , we have к В
42 43 ^0 exact- But с ® & , % е • с?, *» &. . It follows then that the canonical-homomorphism В л. К с & сУ_, -* OL, is injective, i. e. , (*) remains exact when tensored by & , . Consider (R2)-\ 0-(e)- B1 -B-O Tensoring (Cl) by 0%,, (over B1 ) and using the Tor sequence we find that В B! Tor. (©%,• »k) = 0 iff Gf , is B1 flat iff the canonical homomorphism IB В ^в! ®b« ^в1 ** °в' ®в« в' = ^в1 is inJective- <We use the fact B1 Tor. (Ф'-,, i B1 ) = 0. ) Now (C2) is an exact sequence of B-modules, and tensoring 1 В it by O"^ (over B1 ) amounts to tensoring it by 0" (over B). Hence в а (C2) ®_. ©'_, stays exact since & is B-flat. Finally В В В (И) вв, (R2) "в1 11 в, -0 exact , В -0 exact. (R2) ®_., ©*_, is В 13 exact as we observed above. To prove that С_,t is B1 flat, it 15 В is equivalent to proving that a1 is injective, i. e. , Ker a1 = 0. Note that and ~ & в в- Since is B"flat' a в в is injective, and from the diagram it follows that q1 is also injective. Further discussion of Axiom B) а,щЗ*Axiom pj: Suppose we are given . 0-»(?)-* A1 -*A-*0 with rlc(?)=l and a k-algebra homomorphism (p:B-*A. Set В1 = В X A1 . Let us consider the canonical map (*) in Axiom B) of Schles- singer1 s theorem for the functor Def. (X) in more detail, (i. e. , the map Def. (B1 ) -*• Def. (A1 ) xDef ,A) Def- (B))- Suppose we are given deformations (not merely isomorphism classes) X , over B1 , X , over A1 and X В А В over В and isomorphisms Now v# induce isomorphisms 1а , A. But now there is a canonical isomorphism Hence the v# determine an isomorphism (*> By the universal property of the "join11, we get a morphism where Z = Spec(C X _, QT )(X = Spec fr is chosen to be one of the
44 , . ре _> Cf & -* & are then defined objects in (*) and the homomorphisms '^ A' A' A uniquely but the fibre product is independent of the choice for , is in p X : It is the fibre product of 5Г and &^ by A «ъ \ and thus well defined). We claim that f is an isomorphism. From this claim it follows as a consequence that given deformations X and X , of X such . that X_ ®__ A is isomorphic to X , ®A, A, i. e. , given a point Jj В А A |€ (Def. X)(B) XD (Def. X)(A'), a deformation Xfi, which lifts Xfi and X , depends (up to isomorphism) only on the choice of the isomorphism A. X ® A %X ® A, so in particular, we have a surjective map В В А А (*) l8om(X J (Def. X)(A' ) is the canonical ' VB ~B ~ "А1 ~ where \ : (Def. X)(B' ) - (Def. X)(B) *{DeL х)(д) map of Axiom B). ((Def. X)(B) by definition = isomorphism classes of defor- deformations of X, i. e. , all deformations of X , of X over B1 modulo isomorphisms 15 which induce the identity map on X. ) Take the particular case A = k, A1 = к [el, (p: В-A; then X ® A - X, X. , ® . ,* A - X and then the left L J T В В can A' A1 can hand side of (*) consists of a unique element, namely, the one induced by the identity map X -* X. This implies that \ (|) consists of a unique element, and completes the verification of Axiom C). Thus, finally it suffices to prove that the morphism f : X_. -* Z_, is an isomorphism as claimed above. Note 45 that (f ® k) is the identity map X — X and that X_. , Z , are flat over B. В В Hence our claim is a consequence of Lemma 8. 2. Let X = Spec^. and X = Spec^ be two deforma- A A . A A tLons of X = Spec<^. and f : Хл -*X. (f : & . — &K a homomorphism of k- r к А А А А algebras) a morphism such that f 0k is the identity (f ®k is the identity). Then •is an isomorphism. (In the proof, it would suffice to assume & flat/A). Proof. The & can be realized as embedded deformations of ________ j^ X = Spec &., so that we have a diagram Let X1 = f(X ) where X are the variables in PA. We have X1 = X + <P v v v A v v I v where CP « *7^A (X ,. . . ,X ) (Жд = max. ideal of A). It follows easily since 7 v A 1 n A. A is finite over к (as we have seen before) that X и- X1 is just a change of v v coordinates in A , so that 1 is induced by an isomorphism P "•'P.. We can A A A assume without loss of generality that this is the identity. Then it follows that f is induced by an inclusion I. C ** • This implies that f is surjective. Let A A J = Kerf so that r1 -+GrZA -0 is exact. A A Since <У\ is A-flat, it follows that A 0- J k->0 is exact.
47 Since i ®k is the identity, it follows that (J ® k) = 0. Since 7?t is in the radical of О A, by Nakayama1 s lemma, it follows, that J = 0; hence f is an iso- A morphism. This completes the proof of the theorem. §9. The case that X is normal. Let Х^*ЖП be normal of dimension > 2. Let U = X-SingX (Sing X = Singular points of X). We use the well-known Proposition 9.1. Let Z be any smooth (not necessarily affine) scheme over k. Then the set of first order deformations of Z is in one-to-one cor- correspondence with H (Z, 9 ), i.e., (Def. Z)(k[e]) * h\z, 9 ) (9 tangent bundle of Z). Proof. If Z is affine, we have seen that any first order deformation of Z is trivial, i. e. , it is isomorphic to base change by k?e] (this was a con- consequence of the fact that when Ze-* A and Z is smooth, T = @)), Hence any first order deformation of Z is locally base change by k[e]. Hence if {U.} is an affine covering of Z, a first order deformation of Z is given by (p. : (U.PlU.) ® k[c] - (U.PlU.) ® k[c], {(P..} being automorphisms of (U.PlU.)®k[c] satisfying the cocycle condition. It is easy to see that first order deformations correspond to cohomology classes. Now it is easy to see that for an affine scheme W/k, (W = Spec B) AuU ..(W ® к[ф = Derivations of B/k ( = H°(W, в^). The proposition now follows. Lemma 9.1. Let X , X be two deformations of X (not necessarily A A of the first order, A = local, finite over k). Let U1 = X* |U and let @: U -+ U be an isomorphism over A. Then CP extends to an isomorphism (unique) X* - X\ . A A Proof. If X is a deformation of X as above, we shall prove (*) The proposition is an immediate consequence of (*), for <f induces a homo- mo rp his m ,аг , ). XA This implies that CP is induced by a morphism ф ; X -¦ X . It is easily seen * A A that {ф 0 к) is the identity, and this implies easily that ф is an isomorphism. To prove {-^, we note first that if A = k, it is well known. In the general case, we have a representation 0-MO—A-* A —0 exact, = 1.
48 49 By induction it suffices to prove (*) assuming its truth for Then we have 0 - с & •* - О exact, lemma again by an induction as in the previous proposition. This induces an exact sequence of sheaves by flatness 0 -* ^ -¦ —0 with e • & % ft . This is because X is flat over A. As a sheaf its A restriction to U is also exact. Then we get a diagram о - н°(х, ay - н°(хА, &A) - н°(хА , erA ) - о л I Л ° о -* н°(и, ©y - н°(иА, ^A) - h°(ua , <^A ). о о The first and the last vertical arrows are isomorphisms, and it follows then that the middle one is also an isomorphism. This proves the proposition. Remark 9.1. The proposition is equivalent to saying that the morphism of functors (Def. X) - (Def. U) obtained by restriction to U is a monomorphism. Lemma 9. 2. Suppose that X has depth >¦ 3 at all the points (not merely closed points) of Sing X (e.g., dimX>^3 and X is Cohen-Macaulay) (X necessarily normal). Then every deformation U of U extends to a A. deformation XA of X. A Proof. Define X. by X. = Spec <5"v with &. = <Э"„ = H°(UA , <2"A ). A'A^X. AXA AA A A Suppose we have a presentation 0 — (e)-*A-*A -* 0 rlc (e) = 1. We prove this Since depth of &„ at x € Sing X >; 3, by local cohomology the maps X, x are isomorphisms. The isomorphisms iTfX,^ )~ rT(U,^, ) follow from Theorem 3. 8, p. 44, in Hartshorne1 s Local Cohomology. Indeed, we have the following exact sequence 0 — С -* & — 7f-*- 0. We must prove that H Cf) = гГСЗО = О. Theorem 3. 8 (loc. cit. ) states that for X a locally Noetherian prescheme, Y a closed pre- scheme and JS a coherent sheaf on X, the following conditions are equivalent: (i) Н^.(Л) =0, i < n. (ii) depth #]>n. Taking # = & and Y = Sing X, the fact that H°C0 = H C*) = 0 follows immediately. In particular, НГAГ,<У ) = 0. Hence we get (*) 0 - H°(U, Or) - H°(U, (TTT ) - H°(U, e-TT ) - О is exact. Now H°(U, & ) = H°(X, Ck) = &y and H (V, O^ ) = & flat/A by induction hypothesis. Ao
50 . 51 Hence О - С. -+ &. - ©- -0 is exact к о and & flat/A . By Proposition 3.1 it follows that 0* is flat/A А о о and so X represents a deformation of X. A Proposition 9. 2. Let X— АП be such that X is of depth > 3 at \/x € Sing X. Then the morphism of functors Def(X) - Def(U) , U = X-SingX obtained by restriction is an isomorphism, i. e. , (DefX)(A) -* (DefU)(A) is an isomorphism VA local finite over k. Proof. Immediate consequence of the above two lemmas. Remark 9. 2. Suppose Х-АП is normal (with isolated singularities). Then if ex is of depth > 3 at V* « Sing X, X is rigid. For, it suffices to prove that first order deformations of X are trivial and by the preceding to prove this for U. We have an isomorphism Н*(и, 6 ) - ri (X, 6X) because of our hypothesis. But H*(X, 9^ = 0 since X is affine. This implies the assertion. 510. Deformation of a quotient by a finite group action. Theorem 10.1 (Schlessing-e-r--Inventiones f 70). Let Y be.a smooth affine variety over к with char к = 0. Suppose we are given an action of a finite group G on Y such that the isotropy group is trivial except at a finite number of points of Y, so that the normal affine variety X = Y/G has only isolated singularities ( at most the images of the points where the isotropy is not trivial). Then if dimY = dimX>^ 3, X is rigid. (It suffices to assume that the set of points where isotropy is not trivial is of codim >^ 3 in Y. Of course, the singular points of X need not be isolated, but even then X is rigid. ) Proof. Let U = X-SingX, and V = Y-(points at which isotropy is not trivial). Then the canonical morphism V -¦ U is an etale Galois covering with Galois group G. From the foregoing discussion, it suffices to prove that H (U, 6 ) = 0. We have the Cartan-Leray spectral sequence where ^(®у) denotes the G-invariant subsheaf of the direct image of the sheaf of tangent vectors on V. We have 0 = f^ (в ). Hence the above spectral sequence gives the spectral sequence hp(g,h4(v, ey)) => Hp+q(u, ви). Now HP(G,Hq(V, 6y)) = 0 for p>^l since G is finite and the characteristic is zero. Hence the spectral sequence degenerates and we have isom. H°(G, HqtV, в» Hq(U, 0TT).
52 53 In particular, H*(U, 6^ - H°(G, hV, By)). Now 6y is a vector bundle, and since dim Y > 3 and (Y-V) is a finite number of points, 6y is of depth > 3 at every point of (Y-V) (or because codim(Y-V) > 3). Hence rT(V, 9у) = H^Y, в ) = 0. It follows then that H*(U, Q^) = 0, which proves the theorem. Remark 10.1. Let Y be the (x,y)-plane, G = X /2 operating by (x> y) и. (_x> „y)e Then X = Y/G can be identified with the image of the (x, y)- plane in 3 space by the mapping (x, y) i- (x2, xy, у ), so that X can be identified with the cone v = uw in the 3 space (u, v,w). It has an isolated singularity T at the origin, but it is not rigid (cf., the computation of done for the case of a complete intersection, §4 and §6). which has been $11. Deformations oi cones. Let X be a scheme and F a locally free &^~ module of constant rank r. The vector bundle V(F) associated to F is by definition SpecS(F), i. e. , "generalized spec" of S(F)--the sheaf of symmetric algebras of the Ф^ module F. We get a canonical affine morphism p : V(F) - X. The sections of V(F) over X (morphisms s:X-*V(F) such that p« s = Idx can be identified with Homex-alg(S(F), 0^) % Hom^-mod , (F, H°(X, F ), where F* is the dual of F. With this convention, sections of the vectg? bundle^ V(F) considered as a scheme over X are % sections of F over X. Let 1PN+1 - @, . . . , 0,1) ¦? IPN be the "projection of IP onto P from the point @,. . . , 0,1)", i. e. , ir is the morphism obtained by dropping the last coordinate. Let L = IP - @,... y 0,1). Then we see that L has a natural structure of a line bundle over IP , and in fact L^Spec( Ф & N(-n)). n=0 P Proof that L ^ Sp^c( © & xj(-n)): We first remark that in general n=0 PN given F a locally free & -module of constant rank r, the vector bundle V(F) = Spec S(F) associated to F (in the definition of Grothendieck) is such that the geometric points of V(F) correspond to the dual of the vector bundle IF associated to F in the "usual" way, that is, F becomes the sheaf of sections oo of IF. Consequently, in order to prove that L ^Spec( © & M(-n)) we must n=0 IP N show that the invertible sheaf corresponding to L -» P is & N(l)- We do IP this by calculating the transition functions. If we denote the standard covering of P by uQ,. . . , u , then we see for P - @, 0, . . ., 0,1) -* P we have v ((Xq, . . . , X )) is of the form (X , . . . , X , X.) and hence the patching data -1 xi/xi 1 is of the form тг (и.Пи.) ~ С X (u.Plu.) * С X (u. OuJ^T *"" (u.Plu.) (where X. are the standard coordinate functions on P ). This is precisely the patching data for &" P Consequently, QT M(l) is the invertible sheaf associated to PN+1 - @, . . . , 0,1) * IPN. The set of points IPN % S = (*,...,*, 0) С IPN+1 N give a section of L over IP ; we can identify this as the 0-section of the line N+l ^ N+l bundle L. We have a canonical isomorphism Jk, ¦¦ (IP -S) sending @, • . . , 0) ^ @,. . . , 0,1), and hence L-(O-section) - A.N+1-@, . . . , 0). Further- more, it is easily seen that L-(O-section) = Spec( Ф fir „ — ¦" ¦ ^ -oo IP and it is the
54 55 —1 ( bundle associated to L. Suppose now that Y is a closed sub scheme of IP . Let L = x (Y). Then Ь„ is a line bundle over Y and from the preceding we have 00 L =S?ec( О СГ (-a)). 1 n=0 x Let С be the cone over Y, i. e. , if p is the canonical morphism p : (A -@,. . . , 0)) — TP induced by the isomorphism A — (IP -S) defined above, we define C1 = p (Y) and then C1 = C-@,. . . ,0). The point @,. .., 0) is the vertex of the cone С So, С = closure of C1 in A before we have As -@-section) = Spec( П=-00 — N+l N+l Let С be the closure of С in IP (C being identified in IP as above). Then we see that С = LyU @,..., 0,1) , С = Ly - @-sectioxi). We call @, 0,.. . , 1) the vertex of the protective cone C, for @,. .., 0,1) N+l N+l goes to the vertex of С under the canonical isomorphism (IP -S) -* A, If Y is smooth, С is smooth at every point except (possibly) at the vertex. N Consider тг : L -* IP . Let T denote the line bundle on L -consisting N of tangent vectors tangent to the fibres of L -» ]P . Then we have (*) T - 0T - tt*0 -* 0 exact. L N я- : L — Y be the canonical morphism. Then we have a similar exact sequence. Let T denote the bundle of tangents along the fibres of L — Y, so that we have a commutative diagram 0 0 0 0—T (A) j i Г —-—•¦ 6 Y L, I i o-tL - 0 N N+l where N = normal bundle for the immersion L *-• JP , N = normal L Y Y N bundle for the immersion Y^ IP . We have T 2; Т| from which it Y Ly follows that l. e. , CokerFL -©LIL- )- Сокег[(тгу(ву))-*т (^L)L. ] , N %w*(N ). LY Y Let U = C-(vertex). Then if N is the normal bundle for U** AN+1, it is * immediate that N = j (N ) where j : U -* L is the canonical inclusion. U L Y If we denote by the same ж the canonical morphism л* : U -* Y, we deduce that N = тг (N ). Then restricting the bundles in (A) to U we get Let us now suppose that Y is a smooth closed subscheme of IP . Let
57 (Bi) (вг> о -' (В) I О ir*(Ny) The exact sequence (B2) is obtained by restriction to U of the exact sequence H°(A ,© N .) is a free module M over P = k[z ,...,Z ] with basis A -— az JN+1J . We see easily that h is defined by restriction to A -@) of ZN+1 : P-M Cf: where Cj>(l) = 2z. -r— A is generator of P over P). Hence Cf> is a graded i ¦homomorphism, and is therefore defined by a homomorphism of sheaves on N * JP . We see indeed Cf> = 7Г {q> ), where (**) where T 0-T N+l N+1 is the bundle of tangent vectors tangent to the fibres of lA^-(O) jt : A -@) -» IP . We shall now show that we have {& NU) is defined by homogeneous elements of degree ^ 1 in P considered P as a module over P). From this the assertion (C) follows easily, and we leave these details to the reader. Let J be a coherent <У __ module. Then we have (C) IP IP where the second row is the pull-back ir of the well-known sequence ,* Now it 3* is defined by the sheaf of & ^ modules IP oo on P (the middle term of this exact sequence is the direct sum of & A) JP taken (N+l) times). Let us exam: ine the canonical homomorphism h : Т| -* N+l Let (z ,. . . , z ) be the coordinate of A . Then considered as a sheaf of modules over G & (-n)), so that we find 0 P * °° Jr J  = ® 3^("n) anc* hence HP(L, 7Г 3f) = ® HP(P ,
58 59 Similarly, if 3* is at coherent (^.-module, we get HP(U, (U = C-( and N Let us now suppose that Y is ji smooth closed subvariety of IP (of dimension >^ 1) and that it i^ projectively normal, i_. e_. , С is normal. We have then But since С is normal (of dimension.> 2) Now from (B) we get H°(U, Hence T1 = Cokera = Coker(Imp? H°(U, T*Ny)). Now p is a graded homo- morphism of graded modules over Э H°(Y,CY(n)), namely, the gradings are n=0 From(C). identifying в |тт*ж {СГ N(i)) *We get A -@)tU IP For, в I is a trivial vector bundle and hence this follows from the ^n+11 C fact H°(U,e*TT) = H°(C, CTJ. Now U L» N_ is a reflexive ©'.-module since L ————— ^ TT U N_ = Horn (I/I ,Or ), where I is the defining ideal of С in C ^C C of this it follows that H^U.lty =H°(C,NC). Hence we have AN+1. Because and then Ъу (С), р is also a graded homomorphism. Hence (Imp) is a graded submodule of H°(U, ж (в - From *bis il follows that T has a canonical structure of a graded module over k[z ,..., quotient of the graded module and in fact that it is a H°(Y,Nv(n)). Thus we get Proposition 11.1. Let Y be a smooth projective subvariety of IP N
60 61 (of dimension^ 1) such that the cone С over Y is normal. (we have only to suppose that С is normal at its vertex). Then T has a canonical structure of a graded module over k[jz ,. . ., z Л , in fact it is a quotient of the graded. module Ф H°(Y,Ny(n)). S12. Theorems of Pinkham and Schlessinger on deformations of cones. Theorem 12.1. Let Y be as above, i. e., Y is smooth closed-* IP and dimY>l. Then A) (Pinkham). Suppose that T _ is negatively graded, i. e. , T^(m) = 0, m > 0. Then the functor Hilb(C)-Def(C) 1 . is formally smooth. B) (Schlessinger). Suppose that T_ is concentrated in degree 0. Then we have a canonical functor Hilb(Y)-Def(C), and it is formally smooth. In particular, every deformation of С is a cone. Recall that Hilb(C) is the functor such that for local finite k-algebras A Hilb(C)(A) = <ZClPn+ XA, z closed subscheme, Z is flat over A and Z represents an embedded deformation of С over A}. Proof. A) Given 0 —(e) —A1 -* A-* 0 exact with rk(e) = 1 and A1 , A finite local over k, we have to prove that the canonical map (*) Hilb(C)(A' ) - Def(C)(A» ) *Def(c)(A) Hilb(C)(A) is surjective. Take the case A1 =k[c], A = k. In particular (*) implies that (a) HUb(C)(k[c]) -Def(C)(k[?]) is surjective, and (b) given i с Hilb(C)(A), let ?* be the canonical image of ? in Def(C)(A). Then, if % can be extended to a deformation of С over A' , then Jtj € ffilb(C)(A' ) such that -r\»— % (the difference between this and (*) above is that we do not insist that r\ *¦ ?). We claim now that (a) and (b) => (*). (In particular, to prove A) it suffices to check (a) and (b). ) Given % as above,the set of all ^ € Hilb(C)(A' ) such that т]*-* ? (provided there exists an r\ such that t] »— ?) has a structure of a principal homogeneous space under Hilb(C)(k[c]) (see Remark 6.1). [This can be proved in away similar to proving that Hilb(C)(k[e]) ~ H°(C,Hom%,Cr~)), where I is the ideal sheaf defining С in IPN+\ cf. Remark 6. 2. J Similarly, all deformations т^ which extend ? form a principal homogeneous space under Def(C)(k[c]), provided there exists one. Hence,if (b) is satisfied and T] is a deformation extending ?, because of (a) there exists in fact r\,t\ *+ ? and tj »¦* "Л* This completes the proof of the above claim. Now Hilb(C)(k[?]) = H°(C,N~). We have С = L U (vertex) and С is normal at its vertex. Hence H°(C,N~) =H°(Ly,NL )= Э H°(Y,Ny(-n)). If T is negatively graded,it follows that the canonical map С Ф HO(Y,Ny(-n))C ® H°(Y,Ny(n)) n=0 I n=-oo
62 63 is surjective. Hence wr. have checked that ffilb(C)(k[?])- (Def C)(k[cl) is surjective. It remains to check (b). Given ? с ffilb(C)(A), suppose tihat ' N+l ? is locally extendable, i. e., given ? : С — IP , (i) ? can be extended A. A. locally to a deformation over A1 at every point of С , and (ii) this exten- A. sion can be embedded in P , , so as to extend ? locally. We observe that it is superfluous to assume (ii), for we have seen that in the affine case (Embedded Def)(X)-* Def(X) is formally smooth. Then we see that to extend ? to an *i с Ш1Ь(С)(А1), we get an obstruction in H (C, N—) (this is an immediate consequence of the fact already observed, that extensions form a principal homogeneous space, cf. Remark 6. 5). We observe that for ? the property (i) is satisfied. Since there is an ^ € Def(C)(A' ) with ^»- ? (? < Def(C)(A), ? -*I), the condition (i) is satisfied for all x с С. But now G is smooth at every point of C-G. In this case we have already remarked before that (i) is satisfied (cf. , Proof of Propo- Proposition 9.1). Since G is normal, С is of depth >^ 2 at its vertex, so that by local cohomology we get We have Now by hypothesis ? € Def(C)(A) can be extended to r\ € Def(C)(A! ); in particular, x| I gives an extension of ? I . Hence the canonical image of this obstruction in H (U, N ) is zero. (We see that as above, extending an embedded deformation of U gives an obstruction in H (U,N ). ) This implies that there is an -p с Hilb(C)(A' ) such that т\*+?. This checks (Ъ), and the proof of A) is now complete. B) Given a deformation ot Y, we get canonically a deformation of U = C-@). To get a canonical functor Hilb(Y) -*• Def(C), it suffices to prove that a de- deformation of U can be extended to a deformation (which is unique by an earlier consideration). If depth of С at its vertex is > 3 this follows by an .earlier result, but we shall prove this without using it. Let A1 0—(e)--A1 -*A —0 be as usual, and let Y N in IP . Then we have an exact sequence of sheaves. • Y be deformations of Y A 0-0- -er -е- -о, е- Y Y Y Y YAf &L. being A1 flat, etc. Similarly for Y = IP . Then we get a commuta- YA! tive diagram H°(JPN, IP A A • H (Y, <ry(n)) - 0 Y., , A ,N )= © H*(Y,N (- Y 4 n=D Y A (n)) 0 (Cl) A'UY j 1 0 (C2) (n)) -0
64 65 where n ^ 0. It is immediate that (Cl) and (C2) are exact. From this com- commutative diagram, by induction on rk A1 , it follows that ° hV*. , & „ (n)) - H°(Y , fr (n)) - 0 A IP*, A' is exact because (Cl) and (C2)'are exact and the first and third rows are exact. Then as an easy exercise it follows that the second row is exact. Then it follows that H°(YAl,<yv , (n))-H°(Y 9 СГ (n)) - 0 is exact. Define A YA, л хА 00 Of = 0 H°(Y ,,C (n)) (resp. & ). Then we get an exact sequence C A Y CA« n=0 с cA, cA -o, Ч.' Again by induction on rkA1 it follows that & is A1 -flat (for, &^ is A A A-flat by induction hypothesis and by, an earlier lemma (Lemma 8.1) this claim follows). Hence Spec ?T provides a deformation of C, i. e. , it CAl extends the deformation of U = C-@) which is given by the deformation Y^, of Y in IP . This proves the required assertion. Since depth of С at its vertex is >. 3 by an earlier result this is the case. Hence we get a canonical functor • Hilb(Y)-*Def(C). From this stage the proof is similar to A) above. As before it suffices to check assertions similar to (a) and (b) above. We see that Hilb(Y)(k[e]) 2Г H°(Y,Ny). We have oo HO(Y,Ny)C Ф H°(Y,NY(n)) n=-oo i The hypothesis that T consists only of degree 0 elements, implies that С H (Y, N ) — T is surjective, i. e. , i С ffilb(Y)(k[c])-* (Def C)(k[e]) is surjective. This checks (a). To check (b), take 0 — {t) — A1 — A — 0 as- usual, |€ Hilb(Y)(A), I^tic (DefC)(A). Suppose we are given л' e (Def C)(A extending t], then we have to find |! € HLlb(Y)(Af ) extending §. Since Y is smooth, the condition for loca] extension over A is satisfied as above, and hence we get an obstruction element X. с H (Y, N ). We have In a similar way, the element t^l defines an obstruction element 1 °° J Ц « H (U, Ny) = © HT(Y,NY(n)). But by hypothesis this bbstruction is zero. 1 1 By functorially and the fact that H (Y, Ny)^ Q H^(Y,N (n)) it follows that -oo X. = 0. This proves the existence of ?» and the theorem is proved.- Examples where the hypothesis ?f the above theorem are satisfied. N Lemma 12.1. Let Y*5- IP be a smooth projective variety such that ГН {Y, (^y(n)) = 0, Vn / 0, n € Z , ©Y(n)) =0, \/n t 0, n €
66 67 and Y is protectively normal. Then tJ, consists only of degree 0 elements (dim Y >, 2, follows from the hypothesis). The hypothesis H^Y, QT (n)) =0, n h 0 implies that TC = [as graded modules modulo elements of degree 0]. To prove this we have only to write the cohomology exact sequence for (B2) of §11 as well as use (C) of §11. To compute Сокег(я- F .. ) -*j (NY)), use the exact sequence о - N|Y)-»*(Ny)-o. Now н\тг*(еу)) = © H^Y, еу(п)). Since by hypothesis hV, 0y(n)) = 0 for -oo n k 0, it follows by writing the cohomology exact sequence for the above, that * * 1 Сокег(тг @ ._ __) — л- (N )) = T has only degree 0 elements. This proves jpN Y Y С the lemma. Lemma 12. 2. Let YC & be a smooth projective variety such that ^ -y(n)) = 0, \/n > 0 =0, Vn> 0. Then T has only elements in degree ? 0. Proof; The proof is the same as for B) above. Remark 12.1. Given a smooth projective Y and an.ample line bundle L on Y the conditions in A) (resp. B)) above are satisfied for the projective embeddings of Y defined by nl., n» 0 if dimY> 1 (resp. dimY.> 2). Exercise 12. L Let Y = 3 collinear points in IP and Y = 3 non- collinear points in P. Take a deformation of Y to Y_, which obviously exists. Show that this deformation cannot о 1 be extended to a deformation of the affine cone С over Y to the affine cone С over Y. О О ¦ JL 1 [For, if such a deformation exists, we see in fact that J a deformation of С to С , closures respectively of С and C- in IP . We have a (finite) morphism^:C-*C which is an isomorphism outside the vertex and not an isomorphism at the vertex. We have This implies that the arithmetic genus of C_ = (arithmetic genos of С +1), but if there existed a deformation, they would be-equal. J $13. 1 g computation for deformations of the cone over a- rational curve in JP . Lemma 13.1. Let be a- connected curve of degree n, not contained in any hyperplane. Then Y = Y-U...UY where Y. are smooth rational curves of degree n. such that n =2n., each T. spans a linear sub- space of JP of dimension n. and the intersectionstof Y. are transversal. Proof. Let Y be an irreducible curve of. degree < n in JP . Then we claim that there is a hyperplane H such that YCH- Choose n distinct
68 points on Y. There exists a hyperplane H passing through these n points. We claim that H contains Y for if H does not contain Y it follows that deg(H • Y) > n, contradicting the fact deg Y < п.'-Щ% then follows that if Y is an irreducible curve of degree r, r < n, then there is a linear subspace H2:JPS in,Fn such that УСН^Б?8. Now let Y be a curve in IP of degree • n- not contained in any hyper- hyperplane. Let Y. A <^ i ? r) be the irreducible components of Y. Then if n. = deg Y., we have n = Sn.. By the foregoing, if H. is the linear subspace of IP generated by Y., then dimH. <^ п.. Since. Y.1 s generate IP , _a fortiori the H.1 s generate IP . Since Y is connected, UH. is also connected, НПн ? 0 We find i. e. , we can write a sequence H ,. . . , H such that Н.Пн. . 0. We find then i. , easily that if L. is the linear subspace generated by H.,. . ., H.., J r J dimL. < n, + . . . + п.. Since n = 2 n. and dimL = n it follows that dimL. = n + . . . + п.. It follows in particular that dimH. = n. and that L.OH. = one point. It remains to prove that Y. is smooth and rational (the assertions about transvereality are immediate by the foregoing), and this is a consequence of the following Lemma 13. 2. Let Y be an irreducible curve in IP of degree n not contained in any hyperplane. Then Y is a smooth rational curve, in fact parametrized by t и- A, t, t ,...,t ). Proof. Let x. be a singular point of Y. Choose n distinct points x ,. . . ,x on Y. Then there is a hyperplane H such that x. € H. Now x cannot be smooth in HOY, {or if it were so it would follow that x is also smooth-on Y. Frdtn this itfollows easily Deg(H • Y) > n which is a contradiction. Hence every point of Y is smooth. To prove the assertion about parametric representation and ratiorality of Y, project Y from a point p in Y into IP (see $11). We get an irreducible curve У С Д3 • °* degree (n-1). Then it is smooth and by- induction hypothesis Y1 ca~n be parametrized as A, t,. . .., t ). The pro- projection can be identified as the mapping obtained by dropping the last coordinate. Hence the parametric form of Y can be taken as A, t,. . . , t ,f(t)) where f(t) is a polynomial. Take the hyperplane H as x =0; then by the hypothesis > that Y is of degree n, it follows that f(t) polynomial of degree n. Then by change of coordinates we see easily; that thef parametric form of Y is t*4l,t,t2,...,.tn>. Lemma 13. 3. Let Y be an irreducible curve in IP of degree n, not contained in any hyperplane, or equivalently, a (smooth) curve parametrized by A, t,. . . , t ). Then Y is projectively normal. Proof. Let L = & ПШ| у ^ is wel1 known that projective normality of Yd IP is equivalent to the fact that the canonical mapping Cfv : Н°AРП, & n(v))-H°(Y,LV) is suxjective. IP This follows from the following Sublemma 13.1. Let X be a normal projective variety. Then X
70 71 is projectively normal, i. e. L = over X) is normal if and only if . " > 0. 1 *>" JP" - Proof. In general we have A = {functions on X-@)} = в H°(X,LV) о v (see discussion in ? 11). Now since L is ample we have H (X, L ) = 0, v < 0, and therefore A = Э H (X, L ). Now suppose X is normal. Then v>0 {functions on X-@)} = {functions on X}. Also, XC^P = A » and using the same Teasoning as before В = {functions on A } = 0 H (JP ,C (v)). v>0 JP But any function on X can be extended to Ж which implies the natural map B^A is surjective. Therefore H°(Jpn, & (v)) — H°(X, L ) is sur- jective Vv >. 0- In general we have H°(JPtt, QT (v)) — H°(X, 1?) is surjective for v JPn large. Also 0Н°(Х,М is normal since X is. Thus в Н°(Х,1/) is v>0 the integral closure of bn v. Hence if we assume H°(JPtt. QT Jy)) - H°(X, Ъ) JPn is surjective Vv >^0, we have that Ф H (X, L ) is normal. This means X is normal. Now returning to the proof of the lemma, Y ~ JP and L ~ & ЛЫ JP so that L 2: & i(nV)- I* is clear that Cf is injective if and only if Y is not contained in any_Jiyperplane. On the other hand, we have A)) = n An)) 1 Hence Q is an isomorphism, i. e. , Y*-* JP is the immersion defined by ociated to С (n). It is seen easily that IP Any)) is surjective. This implies that CP the complete linear system associated to С .(n). It is seen easily that IP SW(H°AP1, & .Wj- IP is surjective for all v, and the lemma is proved. Lemma 13. 4. Let Y be as in the previous lemma. Then h\y, L ) = 0 , v > 0 n+1. H1(Y,eY(v)) = 0 , v > 0 (the conditions in Lemma 12.1 are satisfied). Proof. It is well known that eL_^<r Al) and H*(jp\ О А*)) = 0, Y" JP1 IP1 v >_ 0. This proves the lemma. Proposition 13.1. Let Y be the nonsingular rational curve in JP of degree n parametrized by tn-(l,t, ...,t ). Let С be the cone in Ж over Y and С its closure in IP . Then the functor Hilb(C) - Def(C) is formally smooth. (Note by Hilb(C) we mean the restriction of the usual Hilb to (local alg. fin./k) and defining deforjnation of C.) Proof. This is an immediate consequence of Lemma 12.1. Let R be the complete local ring associated to the versal deformation of. С (with tangent space = Def (C) (k [e ]) and R1 the completion of the local ring at the point of the scheme Hilb(C) corresponding to C. Let p : R -» R1
73 be the k-algebra homomorphism defined by Hilb(C) — Def(C) to define this homomorphism we need not use the representability of Hilb(C), it suffices to note that Hilb(C) also satisfies Schlessinger1 s axioms). From the above proposition it follows that the homomorphism p is formally smooth. Hence we can conclude that R is reduced (resp. integral, etc. ) iff R1 is reduced (integral, etc. ). To study Hilb(C) we note that С is of degree n in P and that it is smooth outside its vertex. Proposition 13. 2. Let Xе-* 3P be a smooth projective surface of degree n in IP , not contained in any hyperplane. Then X specializes to C, jl.?. , there is a 1-parameter flat family connecting X and C. Proof. Let H be the hyperplane at oo, with equation x = 0 oo n+1 (coordinates (x ,...,x )). Let X be any closed subscheme of IP . о n+1 it is easy to see that X "specializes set theoretically1 to the cone over X * H (with vertex @,. . . , 0,1) and base X * H ) ; in fact we define the о oo Then and that ф is indeed a morphism outside the point x = (@,. . . , 0,1), 0). 1 о Let us denote by Qo the morphism (resp. rational for t = 0) Then for all t ? 0, O? is an isomorphism. Let Z1 = (f (X) (scheme theo- theoretic inverse image). We see that Z' is a closed subscheme of IP X (A -x ). о There exists a closed subscheme Z of IP X A which extends Z' aad such that Z is the closure of Zf as point set. It is easy to see that Z = Z1 LJ @,. . . , 0,1) (as sets). Let p : Z -* A denote the canonical morphism. Then Vt с A , t ? 0, p" (t) ~XA and for t = 0, p" @) - cone over X * H (as t oo sets). The map (X ,. . ., x ) — (x ,. . . , x , tx ) (t к 0) define s an auto - О П+1 О П Пт1 morphism of IP , the image of X under this is denoted by X . The scheme X specializes to a scheme X c-* IP (this follows for example by using the fact that Hilb is proper). From the above discussion it follows that X = r i -v v - y v - orm*» nvpr X • H (as point set) as follows. 1-parameter family Xt. Xj - X, Xq - cone over ^ (a p cone over H- X^ (as point sets). It follows f*om this argument that we can This family is obtained as follows: Consider the morphism n+2 v 1 A X A — defined by ((* xn+1), t) ^ <xq x^ t*a+1). We see that it defines rational morphism choose Z so that p : Z -* A is flat (and then Z is uniquely determined). Suppose now that X is smooth, X • H is smooth and that the cone 00 отег Н • X (scheme theoretic intersection) is normal. Then we will show oo that p @) = X = (cone over X • H ) scheme theoretically. For this, let о oo I be the homogeneous ideal in к [x ,. . . , x Л of all polynomials vanishing on X. Let I = (f ,. . . ,f ) where f. = f.(x , . . . , x .) are homogeneous poly- i q l l о n+1 aornials. Then Xt = V(Ifc) where Jt = (^(хо» • • • »^n, txn+1)) for t к 0. Let
74 75 X'o=V(I'o) where I"q is the ideal ^ = (f^, . . . ,xq, 0)). Let Iq be the ideal of X . Then clearly I! С IQ- Also it is easy to see X^ is the scheme theoretic cone over X! • H^. This means (X^red = (XQ)red. For, since I1 Г I we have Xf I)X . But by assumption X! is normal hence in parti- o о о о о cula* reduced. Therefore X1 = Xq as required. Let us now return to our particular case, where X is a smooth pro- projective surface in РП+1 of degree n not contained in any hyperplane. Then we can choose a suitable hyperplane and call it H^ such that H^ • X is smooth (Bertini) of degree n in 1РП, and not contained in any hyperplane. (The ideal defined by ХПН is C__(-l). We have is exact. This gives 0 *Н°(Х,ё-х) Н°(СГХA)) Н surjection I I injective +1) Pn+1 IP U» exact exact- The second row is exact, and the first vertical arrow is a surjection. The second vertical line is injective. This implies (via diagram chasing) thatthe third^ arrow is an injection which implies ХПН^ is not contained in any hyperplane. (We make this fuss because a priori degree does not imply having the same Hilbert polynomial. )) Then by the preceding lemmas it follows that H^ • X is a rational curve in 1РП parametrized by A, t, . . . , tn) and then by the above discussion it follows that there is a 1-parameter flat family of closed subschemes de- deforming X to С (С is the closure in 1РП of the cone associated to X * H ). oo This completes the proof of the proposition. Remark 13.1. Let P be the Hilbert polynomial of C^* IPn+1. Let H denote the Hilbert scheme of all closed subschemes of jpn with Hilbert polynomial P. Then H is known to be projective. The ring associated with Hilb(C) above,is the completion of the local ring of H at the point corresponding to C. Let H be the open subscheme of H of points h с Н such that (a) the associated subscheme X, of IP is smooth and (b) X, is not n n contained in any hyperplane in IP .It is easy to see that (a) and (b) define an open condition; that (a) defines an open condition is well known, and that (b) also defines an open condition is easily checked. The foregoing shows that as a point set H corresponds to the set of smooth surfaces in 1РП of degree n not contained in any hyperplane and further H contains points associated to projective cones over smooth irreducible rational curves of degree n in IP . The varieties in IP corresponding to points of H have been classified by the following Theorem 13.1 (Nagata). Let Xе-* jpn+ be a closed smooth (irreducible) surface of degree n, not contained in any hyperplane. Then either (a) X is a rational scroll, i. e. , it is a ruled surface where the rulings are lines in P and it is P bundle over P ; in fact X % F .; where F we denote n- 2 r a IP bundle over IP with a section В having self-intersection (B) = -r,
76 77 and then it is embedded by the linear system | B+(n-l)L|, where L denotes the line bundle corresponding to the ruling, or {ti) n = 4 and X = IP given by the Veronese embedding of IP in IP , i. e. the embedding defined by & IP B) v v 4 b or (c) n = 1, X = IP . Remark 13. 2. Let X = IP , and take the Veronese embedding, with -4v*°."i =XlX2' V1=X 2V Then X is a determinantal variety, defined by BX2) minors of the matrix U V, V. о 2 1 V, U. V 2 1 о v, v и. The cone С over the rational quartic curve has a determinantal representa- representation defined, for instance, by letting U specialize to V , i. e., substituting V for U? in the above matrix. Remark 13. 1 The general scroll can be checked to be theydetermin- antal variety defined by BX2) minors of the B X n) matrix x .. . (x +x ) 0 n-1 n+1' X. ... X 1 n and the cone С is obtained by setting x = 0 in this matrix. n+1 Let us now take the case n = 4. Any two scrolls (resp. Veronese surfaces) in IP are equivalent under the projective group. Hence H split. up into two orbits under the projective group: say, H = К UK . We note S x Ci also that there exists no flat family of closed subschemes of IP connecting a scroll and a Veronese. For if there existed one, the Veronese would be topologically isomorphic to a scroll (say, we are over <C). This is not the case: One can see this,for example,by the fact that H (scroll, 2?) = Z , H (IP , Z) = Ж. Since K. are orbits under PGLE), they are locally closed in H , and the nonexistence of a flat family connecting a member of К to К shows that K. (i = 1, 2) are in fact open in H , so that H is the union Cm \ S S of disjoint open sets K_ and К ; in particular, H is reducible. Let % be the point of H determined by С [we fix a particular C, namely, the cone in IP with vertex @,. . . , 0,1) and base the rational curve in IP paramet- parametrized by A, t, t ,. . . , t )J . We saw above that the closure of H contains ?. This implies that H is reducible at ?. We shall now show that any "generalization11 of С is again a projective cone over a smooth rational curve in IP not contained in any hyperplane. In other words, consider a 1-parameter flat family of closed subschemes whose special member is С and whose generic member is W. Then W has only isolated (normal) singularities, is of degree n and not contained in any hyper- hyperplane. We shall how prove more generally that a surface W in IP (which is not smooth) having these properties is a cone of the type C. . To prove this let б be a singular point of W. Then we can find a hyperplane L through 9
79 such that (L • W) is smooth outside в (Bertini1 s theorem). Now L%IP , L • W is connected, (L • W) is not contained in any hyperplane (same argu- argument as for XPlH in the proof of Proposition 13. 2), and it is of degree n. Since it has only one singular point , it follows by Lemmas 13. 1 and 13. 2 that (L • W) consists of n lines meeting at в. This happens for almost all hyperplanes L passing through 9. We take coordinates in IP so that в = @,. . . , 0,1) and take projection on the hyperplane M = {x = 0}. We can suppose that the choice of L is so made that L • W is smooth and not contained in any hyperplane, so that L • W is the rational curve parametrized by A, t, ...,t ) with respect tQ suitable coordinates in L%IPn. Let В be the projected variety in M. It follows that almost all the lines joining 9 to points of В are in W from which it follows immediately that in fact all these lines are in W. In particular, we have В = W • M and W is the cone over B. This proves the required assertion. Thus any "generalization" of С is either a cone of the same type as C, a scroll, or a Veronese. It follows then that in the neighborhood of ?, H consists of only (parts) of three orbits under PGLE), namely, К , К and Д, where Д is the orbit under PGLE) determined by С In a neighborhood of ?, K. and K- are patched along Д. H Remark 13. 4. It has been shown by Pinkham that R is reduced, that the irreducible component corresponding to scrolls is of dimension 2 and the one corresponding to Veronese is of dimension 1, that they are smooth, and that they intersect at the unique point corresponding to С having normal crossings at this point. Remark 13. 5. The above argument can be extended to show that SpecR for n>4 is irreducible. Pinkham shows Spec R has an embedded component at the point corresponding to С and outside this it corresponds to scrolls. Remark 13. 6. The reason that for n = 4 we have got two components' is that in this case the cone over the rational quartic is a determinantal variety in two ways, namely, it can be defined by BX2) minors of or of Xo X2 X4 X4 X3 Remark 13. 7. Case n < 4. Exercise: Discuss. It follows that SpecR where R is the complete local ring defined by Def(C) is again reducible and has only two irreducible components.
80 ai Part II. Elkik1 s Theorems on Ajgfebraiaation. §1. Solutions jaf systems of equations. Let A be a commutative noetherian ring with 1 and В a commutative finitely generated A-algebra, i. e., В = A[X , . . . , X ]/ (f ,. . . , f ), F = (f^ . . . , f ). Then finding a solution in A to Т{ц) = 0, i. e. , finding a vector x = (a.,. . . , a ) € A such that f.(x) =0 A < i < q) , is equivalent to finding a section foif Spec В over Spec A. Let J be the Jacobian matrix of the f.1 s defined by J = ЭХГ* *XN dt nJ (q X N) matrix , We recall that at a point z € Spec B^-* &. represented by a prime ideal in A[X ,. . . , X ], Spec В is smooth- over Spec A at z if and only if: There is a subset (a) = (a ,. . . , a ) of A, 2, . . . , q) such that (iy there exists a (pXp) minor M of the (p X N) matrix a. di a. -r—^ such that M Д 0 (mod-p ), and j (ii) (f.,. . « ,f ) and (f ,...,? ) generate the same ideal at z. i q ai °p If the conditions (i) and (ii) are satisfied at z, then Spec В is of relative codimension p in A (i.e., relative to Spec A). Let F. . bte Шэ ideal (f ,. . :,f ). The condition (ii) above is equivalent to the following: There is a g € A[X ,. .. , X ] such that z I V(g) and - (FV (The subscript g means loeelfessation with respect to the multiplicative set generated by g This implies that g (F)C (F. ,) for some r. Conversely suppose given a g€ A[Xr-...ay, such that «'r>CF(e) (i.%. , g с conductor of F ic F JH&. . : F)). or (a) Then at all points z € Spec В such that g(z) k 0, (F) and F. generate the same ideal (since F С F). Hence the condition (ii) above: can be expressed as: (ii) There is an element g с К. = conductorof W in Ъ% v (i. e. , tbe set (a-) (a) of elements g such that . . ) such that g(z) MO. Let Д = ideal generated by the determinants of the (p X p) minors of the (p X N) matrix tgg-). Let H be the ideal in A [x^ . . . , X ] defined by H = 2 К. .Д, , (a) (a) (a> i.e., the ideal generated by tfee ideals {K. .A. .} where Ш ranges over (a) (a) all subsets of A, . . . , q). Then we see that at a point z € Spec 3е-* A, , Spec В is smooth over Spec A <=> H generates the unit ideal at z <=> z / V(H). Hence we conclude * z € Spec В is sjmooth over Spec A if and only if z / V(H)nSpecB.
82 83 §2. Existence of solutions when A is t-adically complete. With the notations as in the above §1, we have Theorem 2.1. Suppose further that A is complete with respect to the (t)-adic topology, (t) = principal ideal generated by t « A (i. e. , A = limA/(t) ) Let I be an ideal in A. Then there is a positive integer q such that whenever F(a) = 0 (mod Л) , n > n , n > r (i. e. , f.(a) = 0 (mod Л), \/1 < i < q, n > n , n > r) and x — — — о tr с H(a) (H(a) is the ideal in A generated by H evaluated at a, or equivalently, the closed subsclieme of Spec A obtained as the inverse image of V(H) by the s N section Spec A — A defined by (a.,. . . , a )), N then we can find (a' ,. . . , a' ) « A such that* F(a' ) = 0 and a' = a (mod tn'ri). Remark 2.1. The condition tr € H(a) implies that V((t)) DV(H(a)). N So the section s : SpecA^- defined by (a ,. . . , a ) does not pass through V(H) except for the points t = 0. Roughly speaking, the above theorem says N that a section s of A over Spec A which is an approximate section of A Spec В over Spec A and not passing through V(H) except over t = 0 can be approximated by a true section of Spec В over Spec A. Proof of Theorem 2.1. A) We claim that it suffices to prove the —i N following: j h« A (represented as a column vector) such that <*> F(a) = J(a)h(mod t2n"rl), and h=0(tn"rl). To prove this claim we use the Taylor expansion F(a-h) = F(a)-J(a)h+O(h ) (error terms quadratic in h). Now h2 с t2(n"r)L We note further that • 2n~ г We have F(a)-J(a)h с t I and hence I F(a)-J(a)hc t2n'2rl, f i. e., (*) implies F(a-h) с t2(n"r)I, and Set a. = a-h, a = a. Then this gives 1 о с t2(*-r>I, Hence by iteration we can find a. € A such that (i) (ii) i F(a.) € XT (n'r)I, i> 0, and Now by (ii), a1 = lima, exists, and a1 = a (mod t " I). Further (i) implies that F(a' ) = 0. This completes the proof of the claim. N B) We claim that it suffices to prove that there is a z € A such that
84 2n trF(a) = J(a)z (mod t I), and For,we see easily that there is an h с A . such that t h = z and h с t I. Then, if J = J denotes the Jacobian for F, we have tr(F(a)-JF(a)h) = 0 (mod Л), h = 0 (mod tn"rl). Set t F = G; then these relations can be expressed as G(a)-J_(a)h = 0 (mod t2ni), and G h =.0 (mod tn"rl). In particular we certainly have (**) G(a)-J_(a)h = 0 (mod t2n"rl)f and G h = 0 (mod tn"rl). These are just the same as the relations (*) as in Step A) above, with F replaced by G. Hence we conclude (as for F) that there is an a' such that G(af ) = 0, and tn"rI a1 = a (mod t" *I). Thus we conclude that there is an a1 such that • trF(a! ) = 0, and a' = a (mod tn"ri). Note that F(a') € tn"ri since F(a' ) = F(a)+JF(a)(a« -a) (mod t2(n"r)I) and F(a) с Л. Thus we have 85 trF(a') = 0, and F(aT ) с tn'rI. We would like to conclude that F(a' ) = 0; this may not be true; however, we have Lemma 2.1. Let T = {a € A|tqa = 0} and let q be an integer such q ^o that T = T +1 = T +2 = .. . ,etc., (note that T С T , for q1 >q and A being noetherian, the sequence T CT _. .. terminates). Then T П (tm)A = @) for m > q . q na-q Proof. Let ас Т ПЛ. Then a = tma' =t °(t °a'). Since Чо П+Чо t a = 0, we have t a' = 0. But by the choice of q , we have in fact q q xn-q m-q q t °a' = 0. Hence a = t °t °af = t °t °a* = 0. This proves the lemma. By the lemma if (n-r) > q , then trF(af ) = 0, and F(a' ) « tn"rl implies that F(af ) = 0. Thus the claim B) is proved. C) Let (P) = (P,,...,|3 ) denote a subset of p elements of A,. . . , N). p Э?<ч Given (a) - (a_, .. . , a ) С (h • • • # Я)> let 6 _ be the pXp minor (-zrz—) I p a, p с А of the Jacobian matrix J. Then we claim that it suffices to prove the following: Given к с K, * and 6 € A F defined as above) then there exists (a) ap a ap a z = 0 (mod t I) such that k(aN A(a)F(a) = G(a)z (mod t2ni). . Forwe have t = S X. _k (aN _(a). Then by hypothesis, given к a ap a ap a afnd|6 ,^г we have, z such that z fl = 0 (mod t I) and (mod Л);
86 87 then we see that if we set z = 2 X. _z . we have tr • F(a) = J(a)z (mod t2ni), and z = 0 (mod t I), which is the claim B). D) We can assume without loss of generality that (a) = A,. . . , p) and F) = A,. . . ,p) so that б _ = detM where ap df. 1 < i ? p Then we have J = Let N be the matrix formed by the determinants of the (p-1) X (p-1) minors of M so that we have MN = б • Id , б = б л , a, p as above. We denote by If the (N X p) matrix by adding zeros to N. Let к € К, ., (a) as above and б be as above. Then we claim that if z is defined by Pal -CT , Z is an (NX 1) matrix; then if z = Z(a), we have k(aN(a)F(a) = J(a)z (mod t2nl), and z = 0 (mod Л). By the foregoing, if we prove this claim, then the proof of the theorem would be completed. .« To prove this claim, we observe that the relation z = 0 (mod t I) is immediate. Since к € К. v, (a) as above, we have (I) This gives kf = Z X f, 1< j<q. 1 i=l 3 d? p Bt = S X .. -r— (mod F), 1 < j < q. 1 i=l lj3Xl Substituting (a), we get Э?. р Ы. k(ah-f(a) = S X.. -~r(a) (mod F(a)) i = S X..(ah~-(a) (mod Л). i=l Ч ЭХ1 Let u с A be an element such that u 5 0 (mod *2я)# where Ъ is some ideal of A. Let v = J(a) • u с AN. Then N 3f k(a)v = k(a) 1=1 N p = S (S 1=1 i=l 3f. N oi. X..(a) ' ( 2 ~-(a)u.) (mod lj 1=1 ЭХ1 l X..(a)v. (mod t 1J x In other words, (И) k(a)v. = 2 \. .(a)v. (mod t lb), for 1 < j < q, J i=l 1J '
88 89 where u € A , тг =¦ J(a)u, and u = 0 TNT Now take for u e A and v the elements -Id Then u = 0 (mod t I) and we have k(a),y. « 2 \..(a>v. (mod t И Moreover, v = J(a) [mn] *nJ L1. :] [r] (a) 6(a)fp(a) Hence v. = 6(a)f.(a), 1 < i < p. By (П), k(a)J(a) = k(aN(a) (modt which is precisely the claim in Step D) above. This completes the proof 6f the theorem as remarked before. Remark 2.1. A better proof of the theorem is along the following lines: Introduce a set of relations for F so that we have an exact sequence (Here F denotes f L qj as well as the ideal generated by f. and P = A[X ,.. . »XN]» and R is the matrix of relations of F; it has entries in P.) In matrix notation, we have F* R = 0. Differentiating with respect to X., we obtain JR = 0 (mod F), 3f. where J is the Jacobian matrix (д ) (it is an N X q matrix and is therefore j the transpose of the matrix J introduced in the theorem above). Let P = P/F and define J, R similarly. Then we obtain a complex By (I), k(ayv, ? ^ X..(a)v. {mod P 1J P 2 Hence it follows that , »o that Tor: P S pq -* p is a complex mod F). It is not difficult to see that at a point x с Spec(P/F), the complex (*) is homotopic to the identity (in particular, is exact), i. e. , denoting by a subscript x the localization at x, there is a } ' diagram p 1 5 p Я J -1 p Я J 1 X X =
90 91 if and only if Spec(P/F) is smooth at x. For example, suppose that Spec(P/F) is smooth at x. Then we can assume without loss of generality that di. 1 < i < p det(-rrr-) , . . . . , is a unit in P . From this it follows easily that there dX. 1 <¦ j <_ p x is a direct summand Qf~~Pq, Q %PP such that ImfJ) = Im(jfQ ) and j|Q- : Q -*Im(J) is an isomorphism (JJQ. denotes the restriction to Q ). Indeed, we can take О to be the submodule generated by the first p-coordinates. n af. (Note we have J(e.) = 2 ^pL, with (e.) a basis of P4, and (?.) a basis 1 i=l dXl l г г ~"N — of P ). We see also that Im(R) is of rank (q-p) and is a direct summand in P , In fact the relations give elements of the form in bnR. Suppose now S $i,I. is a relation; then as we have seen before i=l * X (on relations for a complete intersection), ц. с (f , . . . , f.) so that /i. = 0. Hence we conclude that (bnR) is precisely the submodule generated by P _ _ (e. - 2 X. .e.), j >^ p+1, which shows-that ImR is of rank (q-p) and is a direct summand in P 4. Now we see easily that the .complex (*) jjs homotopic to the identity at x if and only if, (i) (*) is exact and (ii)ImR is a direct summand which admits a complement Q such that J|Q, is an isomorphism and Im J ^ImfJlQ^). Now we have checked these conditions when Spec(P) is smooth at x. Conversely if (i) and (ii) are satisfied (at x), it can be checked that Spec(P) is smooth at x. Suppose more generally that we are given a complex (we keep the same notation). Then we can define an ideal Ifb in P which measures the nonsplitting of the complex as follows: *K *s tne ideal generated by elements h such that there are maps r, j: p1 5: pq J p such that It can be shown that J? is the following ideal: Take a (pXp) minor M in J and a "complementary" (q-p) X (q-p) minor К in R (involving "comple- "complementary indices11); then ft is the ideal generated by the elements (detM)(detK). In our case, SpecP - V(}?) is the open subscheme of smooth points. It follows then that 7t and the ideal H (H is the ideal defined before and H denotes the image of H in В = P/F) have the same radical. In our case we have then Rr+j J = h (mod F), for some h с P with image h in If. Multiplying by F (where F is a vector now), we have FRr+FjJ = Fh (mod F2). Since FR = 0, this gives FjJ = Fh (mod F ). '.t t We have used the transposes of the original J; setting z = j F and taking
92 93 "evaluation at a11, we get h(a)F(a) = J(a)z~(mod t2ni). Now a power of h is contained in H since H and ft have the same radical. This is the crucial step in the proof of the theorem above. Hence from this the proof of the theorem follows easily. Before going to the next theorem, let us recall the following facts about Ot-adiс rings (cf. Serre, Algebre Locale, Chap. П, A). Let (A,01) be a Zariski ring, i.e. , A is anoetherian ring, is an ideal contained in the Jacobson radic RadA of A, and A is endowed with theOL-adic topology, i. e. , a fundamental system of neighborhoods of 0 is formed by ОЬ . Since О Л = @), it follows n that this topology is Hausdorff. Let A denote the<7t-adic completion. Then A is noetherian. If M is an A-module of finite type we can consider the 01-adic topology on M and similarly it is Hausdorff and i/ M denotes the <Jt-adic completion of M, we have M = M 0 A. In fact the functor МкМ is exact. Suppose now that M = M. Then we note that any submodule N of M is closed with respect to the Л-adic topology (for, the quotient topology in M/N is the fft-adic topology and since M/N is of finite type and ObQ RadA, this topology is Hausdortf and hence N is closed). If A = A, i. e., A is complete, then any module of finite type is complete for the adic topology so that we don1 t have to assume further that M = M. Let A = A, M be as usual and t € OL. Then M/tM is complete for the tft/tdt-adic topology, for this is simply the/Tt-adic topology on M/tM. Let A = A and 2*Ъе an ideal in A. Given an r, suppose that the relation СП Qlf + OL , m» 0, holds. Then <Я-ГСЪ» for our relation implies that <гГ(А/МСПЛт(А/Ы. Since A/Z» is Hausdorff for the Ot-adic topology. m it follows that CL А/Ь = @), which implies CL С Ъ* $3. The case of a henselian pair (A,0t). Theorem 3.1. With the same notations for А, В as in the pages preceding Theorem 2.1, suppose further that (A,OL) is a henselian pair (in particula-r CLQ RadA), and that A is the JC-adic completion of A. Suppose we are given a € A such that F(a) =0 (i. e. , a formal solution) and CL * A (or briefly Ob )C H(a) for some r. (This means that the section of Spec В over Spec A where B=A|x|/(f.) represented by a passes through smooth points of the morphism Spec В -* Spec A except over V{av). ) Then for all N n >. 1 (or equivalently for all n sufficiently large) there is an a € A such that F(a) = 0, and a = a (mod a~L ). Proof. il) Reduction t? the case ОС principal. Let CL = (t ,. . ., V). Let us try to prove the theorem by induction on k. If «k = 0, then the theorem is trivial. So assume the theorem proved for (k-1). We observe that the couple (A/t , (t ,. . . ,t )) is again a henselian pair Vi >1 (here t , ...,t denote the canonical images of t. in A/t ). Definition. By a henselian pair we mean a ring A and an ideal 0"CC RadA ( = Jacobson radical of A) such that given F = (f ,. . . , f ), N elements of A[Xr . . . , XN] and x° € АП, x° = (x? . .., x^) such that F(x°) = OmodfcrU and such that det( ) 1 is invertible modffl», then HxcA ,x = x mod@L) ax. |xo with F(x) = 0.
94 95 Set t = t. and A = A/(t ). We note also that the OTL-adic topology on A is tHe same as the (t ,. . . , t .) = fft -adic topology. If A, A. denote the cor- corresponding completions, we get a canonical surjective homomorphism A-*A whose kernel is t *A. Let b be the canonical image of a in A . Then we have F(b) = 0. Besides,we see that Ot С H(b) for some s (this follows from the fact that V(H)PlSpec В = locus of nonsmooth points for Spec В -• Spec A and the set of smooth points behaves well by base change and for us the base change is A — A . We cannot say that s - r, for the ideal H (or rather H(modB)) which we have defined using the base ring A does not behave well with respect to base change. The ideal 7 does behave well with respect to base change, and if we had used this ideal we could have got the same integers). Hence by induction hypothesis, for all m^ t, there exists a b с A™ such that F(b) =0, and b = b (mod<7tm). N Lift b to an element а. с A and choose / so that f >_ m. Then we see that a = a (modtfL ), and F(ax) =0 (mocKt1)). (The fact that b = b (mod<7Lm) implies (a -a) + x с <Я,Ш with x € Ker(A - A ) = (t ). Now t € CTL, and if i >; m, (t ) € at. ) We claim that if m » 0 (and consequently / » 0), we have СП QH{oi ). By hypothesis we have OL С H(a), and from the relation a = a (mod 0ЪШ), we get (as ideals in A; to deduce this we use the Taylor expansion). Then, as we remarked before the theorem for m*>> 0, this implies that Ob Q Ща^. Since t с at it follows that t € H(a ). Let A denote the t-adic completion. Then the following -relations in A, for all / > 0, = 0 (mod(t )) N г there exists a € A such that t € hold a fortiori in A , and hence by Theorem 2.1 we can find a1 A such that — ¦ t F(a! ) = 0, and Note that the pair (A, (t)) is.also henselian. Hence if the theorem were true . for к = 1, we would have \/n, such that a) = °» and a = a1 (mod tnA). But we have a1 = a (moid(tS)At), for s sufficiently large, and a = a @L.m- A), m sufficiently large. These imply a=I(mod<Jon) (since t € cjc and AtC A), which implies the theorem. Hence we have^only to prove the theorem in the case к = 1, i- e. , OL principal. B) A general lemma: Lemma 3. 2. Let А, В be as in the pages preceding the theorem, i.e., В = A[Xr ...,XN]/(fr...,f ), F = (fj,...,^). Let С be the symmetric
96 97 algebra on F/F over B, so that Spec С is the conormal bundle' over Spec В (F/F as a B-module is the conormal sheaf over Spec B). Let f, g, h denote the canonical morphisms Spec С — Spec В — Spec A, g = h - f. f h Let V be the open subscheme of Spec В where Spec В -» Spec A is smooth and V1 = f" (V). Then we have the following: (a) g : Spec С-* Spec A is smooth and of relative dimension N (over A) on V1 and (b) J an imbedding Spec С*— Ж q (A-morphism) such that the A restriction of the normal sheaf (for this imbedding) to every affine open subset U^* Vf is trivial. Proof of Lemma 3. 2. We set ,I), К = On V we have the exact sequence (i) O-F/F - (This is an abuse of notation; strictly speaking we have to write (F/F ) I . . . , etc. ) On V we have the exact sequence (ii) 0-*f nc/B*f*(F/F2). Taking f^* of the first sequence,we get sequences 0 - f*(F/F2) - (Free) - - 0 which are exact on any affine open set U in V1 . These exact sequences are split on U, so that we conclude fl i is free on U. On V we have the exact sequence c-oc/A-o. Thus it follows that on U K/K2 Q (Free) = (Free), of rank N (From the exact sequence (ii) it follows that &C#A is °* rank N over С and this implies the assertion (a). ) If we introduce N more indeterminates Zr...,ZN, then •(*) С = A[X, Y, Zr . . . , Zj/(K, Z^ Let K1 = (K, Z ,. . ., Z ). Then we see easily that ; К1 /К1 2 = К/К2 Ф (Free of rkN). It follows that K1/K1 is free on U. Thus fo.r the embedding Spec O*A-A q, the restriction of the normal bundle К1 /К1 to U is trivial. This completes the proof of the lemma.
98 99 C) We. saw in A) above ,that for the theorem it suffices to prove it in the case oc= (t). The condition Ся/лСШа) becomes t* € H(a). (Note that H(a) is the ideal in A generated by evaluating at a elements of H and H is an ideal in A^, ...,XN] not in A-fX^ . . . , 3^]. ) "We' claim now that there jls_ an h € H such that h(a) = (unit) • tr. A priori it is clear there is an h € H'A[x] such that h(a) = tr. Since A is the t-adir completion of A, we can find an h с Н such that h = h (mod(tr+1)) (i. e., the coefficients of h and h differ respectively by an element of (tr+ )). This implies that h(aL) - h(a) = *tr+\ hence h(I) = tr(l+*t). Now (l+*t) is a unit in A. This proves the required claim. D) The final step. Since Spec С is a vector bundle over Spec B, we have the 0-section Spec С*?Spec B. We are given I € AN such that FG) = 0. Now a determines a section of Spec(B ® A) over A, or equivalently a morphism A s : SpecA -» Spec В forming a commutative diagram Spec A -*- Spec В SpecA Using the 0-section, s can be lifted to a morphism s : SpecA — Spec C: ~ Sl SpecA — Spec С \ SpecA Now By C), s carries (SpecA-V(t)) to Spec С [l/h], where h is as in C) (here h denotes the canonical image in С of the h in C)). NWe observe that Spec С [l/h] С V1, (V1 = f (V), V = locus of smooth points of Spec B —SpecA) and V1 is contained in the locus of smooth points of the map Spec С -* SpecA. We have С = A[X, Y, Z] /K1 where K1 = (F, I, Z ,. .. , Z ). The section s defines a solution K1 (a1 ) = 0, where а1 с А Ч (a1 extends the section 7). We have t с H(a! ) for a suitable k. Hence the conditions,similar to those of Spec В — SpecA, are now satisfied for the map Spec С — SpecA. It is immediately seen that it suffices to solve for the case Spec С — SpecA, in fact if we get a solution .- a1 « A , we have to take for a the first N coordinates of a1 . Let U = Spec С [l/h]. Then the restriction of the normal bundle of the imbedding Spec О- А. Ч is trivial on U and, U being smooth over A, , it follows easily that U is open in a global complete intersection. By this we mean there exist g ,. . . , g € A [X, Y, Z] such that V(g ,. . . , g ) has dimension N and we have an open immersion (SpecC[l/t]) = SpecC[l/h]^*SpecA[x,Y, Let G = (g1,. . . , gN+ ). Then we have K;=ct. n+q'
100 101 (We denote the localization with respect to t by a subscript. Note that localizationwith respect to t is the same as localization with respect to b. ) The given solution a' in A q is such that K' (a' ) gives rise to a solution G(a' ) = 0 by changing the g., multiplying them by suitable powers of t. Conversely, suppose we have solved the problem for G, i. e. , we have found a' € A such that G(a ) = 0 and a' = a (mod t ). Then we see easily that there is a 8 such that t9 • F(a' ) = 0. Since F(a) =0 by Taylor expansion, it follows that F(a' ) = 0 (mod t ). Now if n» 0, by Lemma 2.1, it follows that F(a' ) = 0. Thus it suffices to solve the problem for G, i; e. , for the morphism Spec С -SpecА, С = A[X, Y, Z]/G. We have seen that a1 defines a section of this having the required properties. Further, Spec C1 is a global complete intersection and smooth over A in* A , i. e. , we have reduced the theorem to the following lemma. §4. Tougeron1 j5 lemma. Lemma 4.1. Let (A,Ob) be a henselian pair and f. € aTy., . . . , Y__] , Э? l N К i < m. Let J = (т~) be the Jacobian matrix, 1 < i < m, 1 < j < N. — — Ox — — — — (т) Ox. J Suppose we are given у = (у ,. . . >У^) « A such that f(Y°) ?0 (mod Д2Ь) where V(k) = V(tft) (or <=> Ъ is also a defining ideal for (A, G1)) and Д is the annihilator of the A-module С presented by the relation matrix (i. e., С is the cokernel of the homomorphism A -* A whose matrix is N J(y )). Then there is а у с A such that f(y) = 0 and у = y° (mod Д^). Proof. The henselian property of (A,?7L) is used in the following manner: Let F = (F-,... , F ) be N elements of A[Y., . ¦., Y ] and У = (Yj # • • • # YN) « A such that (P) (i) F(y") = 0 (modOl) ar (ii) detfr^-) is a unit (mod OL), oY, о J У=У Then there is а у € A such that F(y) = 0 and у = y° (mod Ob). Let 5.,..., 6 generate the annihilator of A. This implies that there exist N X m matrices such that Write m components. We try to solve the equations f(y° + 2 6.U.) = 0 for elements U. = (U ,. . ., U ) « A (we consider vectors in A to be
J02 103 column matrices). Expansion by Taylor1 s formula in vector notation gives 0 = f(y°) + J- B6.U.) + 2 6.6.Q.., ii uj i J iJ where and Expanding, we get J is an (m X N) matrix, f(y ) is an (m X 1) matrix, U. is an (N X 1) matrix (not A X N) matrix as it is written), Q..,?.. are (m X1) matrices. where M is an Nr X Nr matrix of polynomials in Z and every m has no constant term. Let z° с A represent the vector @,... ,0); then we have F(z ) = 0 (modi?) since с..у с Ъ . Without loss of generality we can suppose 1j -CL since }j is also a defining ideal for (A, CC) F(Z°) =0 (modJL). 0 = J-(S 6.U.) + 2 6.6. (Q. .+?..), isli i ui i J ч и (m XN)(N XI) matrix (m XI) matrix = 2 6.(JU.) + 2 6. • JN. i=l i j = 2 F.J) • U. + 26.JBN.(Q. .+?..)) i=1 i i . i j J ч U Thus it suffices to solve the r equations (*) 0 = U + SN^Q^+ j J iJ F .Id = JN.) F. are scalars). К i < r. This is an equation for an (N XI) matrix. Thus (*) gives Nr equations in the Nr unknowns U. , 1 < i < r, 1 < v < N. lV _ _ __ _ We note that Q.. are vectors of polynomials in U. all of whose terms are of degree >. 2. Let F = F ,. . . , FNR 6 A[uiv] represent the right hand side of (*). Write Z, ...,Z for the indeterminate s U.y. Then , M = (m J , (Nr X Nr) matrix ap k Further, we have (^_ )_ o is a unit in A/OL. Hence by the henselian property of {A,oL)9 w^e have a solution z of (*) in A such that z = z (mod GL), i. e., z = 0 (mod OL) since z = @). Set о у = у +z. Then we have о у = у (mod A0L) and f(у) = 0, which proves the lemma. ; Remark 4.1. It is possible to take Ъ such that Ъ Q.OL ', for if (A,(Jl) is a henselian pair, the henselian property is true for 2p , & < Then the above proof also goes through for this case. Corollary. Let (A,0L) be a henselian pair and and CD the canonical mo rphism Cp : Spec A [yr . .., , fm> - Spec A
104 Suppose that Cf is smooth and a relative complete intersection. Then given у € A ((Jt-adic completion of. A) such that f (У) = 0, there is а у с A such that f(y) = 0 and у = у (mod (Я* for any given с >^ 1. Proof. We can take с = 1 for Ot is also a defining ideal for (A,crc). We can a^ fortiori find у « A such that f(y°) = 0 (mod ОЦ { <= f(y) = 0).. y° then defines В a section of Spec(B) over SpecA/ot. The morphism is a complete intersection and smooth at points of this section. This implies that the Ideal generated by canonical images in A/Ob of the determinants of the (m X m) minors of J(y ) is the unit ideal in A/OC, i. e., the canonical о image of Д in A/<7L is the.unit ideal. Since OX is in the Jacobsoa radical, it follows that Д is itself the unit ideal. Indeed Д being the unit ideal in A/tft implies there exists a u « Д such that u = 1 (mod0L), hence u-l с OL, hence u = 1+r, г с 0L. Since OLQUadA, u is a unit in A. §5. Existence of algebraic deformations of isolated singularities. Definition 5.1. A family of isolated singularities is a scheme X^S over S = Spec A, A a k-algebra such that (i) v is flat, of finite pres- presentation and X is affine, X = Spec & and (ii) if Г is the closed subset of X where ir is not smooth, then Г — S is a finite morphism (for this let us say that Г is endowed with the canonical structure of a reduced scheme). -\ Definition 5. 2. We say that two families X — S and X1 — S of isolated singularities are equivalent or is ото гр hi с if there is a family of isolated singularities X"-* S (note that S is the same) and etale morphisms X'UX', X"-X such that (i) the following diagram is commutative Y» etale у j\etale X X1 and (ii) these maps induce isomorphisms Г Г1 . An equivalence class of isolated singularities represented by X -*- S is therefore the henselization of X along Г. r. Consider in particular a one point "family11 X -* Speck with an isolated singularity. (We could also take a finite number of isolated singu- singularities. ) We see easily that the formal deformation space of X (in the sense о of Schlessinger defined before) depends only on the equivalence class of X . Let A be the formal versal deformation space associated to X (we can . speak of the versal deformation space of X by taking Zariski tangent space of A = dimT ), i. e., A is a complete local ring, and we are given a sequence :{X } of deformations over A * n n n+1 X = Spec Cf , A = A/m" , Q- ® A , - n r n n A ' ^n n-1 = •r satisfying the versal property mentioned before. Note that by a versal
106 107 deformation it is not meant that there is a deformation of Xq over A. The following theorem,proved by Elkik, asserts that in fact a deformation of Xq over A does exist (i. e. , in the case of isolated singularities). It is the crucial step for the existence of "algebraic" versal deformations for X~. Theorem 5.1. Let X = Spec <У , & = k[X ,. . . , X ] /(f), о о о 1 m (f) = if ,. . . ,f ). Let X = Spec & be as above, but we suppose moreover that X is equidimensional of dimension d. Then there is a deformation X1 over A such that Xf ® A ^ & , and if X1 = Spec &x then 6" is an n n A-algebra of finite type. (We do not claim that &x has the same presentation as <3T .) о Proof. Let С-ИтО", & = A [x . . , X ]/(f(n)), where - «- n n n1- 1 mJ ffn) с A [X] is a lifting of f. € k[x]. Let A[x]A denote the m-adic (m = m ) completion of A[x]. We see that A[x] is the set of formal power A series Sa.X*1' such that a. - 0 in the m-adic topology of A. Then 7. = limf. is in A[x] and we see easily that & =A[X]A/G). (For, we see that we have a canonical homomorphism o:A[X]A/(f)-5> obtained from the canonical homomorphism A[x]A/(f) —A [x]/(f ). It is easy to see that a is an isomorphism. ) The proof of the theorem is divided into the following steps: A) It is enough to find a (flat) deformation Xм over A of Xq such that O" 8. А, ^вГл, where X" = Speed'" (recall that A. = A/m ). For, given a (flat) deformation X" over A we get deformations {X*1} = Xn ® A over A . For each n, we get then by the versal property of A, a homomorphism a : A-A . n n Note that a is defined by n (a ) : A — A , m » 0, n m m n so that X ® A ^X!l. These {a } are consistent and hence {a } patch m A n n n n m up to define a homomorphism of rings a : A-A. The hypothesis that ©"' ® A « & ' implies that A 1 1 a = Id (mod m ). A. This condition on a implies that a is an isomorphism; for it follows that a induces an isomorphism on the Zariski tangent spaces, so that Ima contains a set of generators of m , hence a is surjective; further, this condition * A implies that the induced homomorphisms a : A/m -¦ A/m are surjective, n A. A and these vector spaces being finite-dimensional, it follows that a is an n isomorphism for all n (in particular injective). It follows easily that KeraC ^mA = №)» i. e. , a is injective. Hence a is an isomorphism, n Now define the deformation X! over A as the pull back of Xй over A by the isomorphism a-1. It is easily checked that X1 8>A A ^;X , and An n this proves A). _ Let us set X = SpecC. Consider the Jacobian matrix J = эх
108 109 l<.i?r, l<J<m ((f) = (f ,.. . ,f )). Let Г denote the locus of points in X = Spec С where rk J < (m-d). Then Г is a closed subscheme in X. (We эт aote that f.« А[х]л and -ттр « А[х]л so that the Jacobian matrix J is a matrix of elements in A[x]A.) Hence if x € SpecA[x]A (in particular if x € Spec©" = Xе-* Spec A [x]), we can talk of the rank of J at x, i. e. , the matrix J(x) whose elements are the canonical images in k(x) (residue field at x) of the elements of J. It follows then easily that the locus of points x of X where J(x) is of rank < (m-d) is closed in X; in fact we see that Г = V(I) where I is the ideal generated by the determinants of all the (m-d) X (m-d) minors of J1 where J1 is J with elements replaced by their canonical images in Cf. It is clear tjiat ГПХ is precisely the set о of singular points of X , which is by our hypothesis a finite subset of X . It can then be seen without much difficulty that Г is finite over А (Г is endowed with the canonical structure of a reduced scheme or a scheme structure from the ideal I introduced above). The proof of this is similar to the fact:quasifinite implies finite in the "formal case11, i. e. , in the situation A -* В where А, В are complete local rings and В is the completion of a local ring of an A - algebra of finite type. Let Г = Spec ©7 Д and let A be tne ideal in & defined by A о о (i. e. , the canonical image of A 0 к in СГ ). By the No ether Normalization lemma we can find y. ,. . . , у in A such that СГ is a finite kfy, ,. . . , у ,1 1 а о о 1 a. module such that the set of comiaon zeros of y. is precisely the set of singular points of X . Lift y. to elements y_,. . . ,y• in A so that y_,.. . ,y vanish on Г. Then we have 1 d B) СГ is a finite А [у]л module (and Г is precisely the Iqcus of y. =0). This is again obtained by an argument generalizing^quasifinite implies finite in the "formal11 cases. " C) The open subscheme Х-Г of X is regular (over А) (ь^е., Х-Г is flat over A and the fibres are regular). This is a generalization of the Jacobian criterion of regularityto the adic and formal case. D) Outside'the set {y = 0} in SpecA[y]A (this is a section of SpecA[y]A over SpejcA and Г lies over this set), Cf is locally free over A[y]A say of rank r, i._e. , pj& ) is a locally free sheaf of ^SpecA Гу"|Л-{у -о} X —1 1 modules (p : X ~* Spec A [y] A canonical morphism). For, p^O" ) is finite over SpecA[y]A. Now SpecA[y]A is regular X over Spec A. We have seen that Х-Г is regular over SpecA, so that it is in particular Cohen-Macaulay over SpecA. Now a Cohen-Macaulay module :M (of finite type) over a regular local ring В is free (cf. Serre1 s "Algebre I locale") and from this D) follows. (We can use this property only for the •corresponding fibres, but then the required property is an easy consequence of this. ) E) Set P=A[y]A. Then for & considered as a P module we have a rep- resentation of the form Am ir An — л p —i»p — &-~ 0 (exact as P modules) with rk(a..) <^ (n-r) (i. e. , determinants of all minors of rk(n-r+l) of (a..) are zero).
110 111 More generally, let us try to describe an R-algebra В having a rep- representation of the form (a .) Г R —¦="*¦ R -» В -* 0 exact sequence (*) < of R-modules a..)< (n-r). Then we have Lemma 5.1. J an affine scheme V (of finite type) over Spec Ж such that every R-algebra of the form (*) is induced by an R-valued point of V. Proof of Lemma. To each R we consider the functor _ Г set of all commutative R-algebras B'l " J^with a representation of the form (#) J . One would like to represent the functor *3* by an affine scheme, etc. We don11 succeed in doing this, but we will represent a functor G such that we have a surjective morphism G -¦ 3^ An algebra structure on В is given by an R-homomorphism В Ф_ В -* В. XV. Then in the diagram 31 the homomorphism R R ® R R R — R factors via a homomorphism R ® R -* R (of course not uniquely deterinined). Now the following sequence| (R Rn) 9 (Rn в Rm) t Rn в Rn -. в 0 в - 0 is exact where ф = (a..) 0 Id Q Id ® (a..). [We have (Ker(Rn-* B) ® Rn) В (Rn ® (KerRn- В)) °аП ОП^ Ker(Rn ® Rn - В ® В) - 0. This implies exactness of the given diagram.] Again there exists a lifting of the above commutative diagram (RmeRn)e<RneRm)^RneRn-B®B B-0 XV. •0 . R 1 „ R * В On the other hand, giving a commutative diagram (Rm в R°) О (R° в Rm) t Rn S Rn Vм where ф - (a..) (8> Id Q Id ® (a..) determines the commutative diagram (I ). The algebra structure on В induced by (I ) is associative if the diagram Id»b\ and the map |ЬвИ1я° factorizes through Rm-Rn, i., (U) or, b» (Id в b) - b- (b ® Id) is zero in B.
112 113 Let b:RnXRn-*Rn be the homomorphism obtained by changing b : Rn X Rn — Rn by the involution defined by x®y^y ®x in R 0 R. Then the algebra structure is commutative if there is a homomorphism 6 : Rn ®та Rn :* Rm such that R such that the following diagram is commutative: (Rm В Rn) О (Rn (I) -R" К. (Ш) commutes. Finally the identity element 1 « В can be lifted to an element e « R (determines a homomorphism R — R ) and there is a map e : R -* R such n e®b-Id n —R (IV) а..) commutes, i. e. , e ® b-Id = 0 in B. Let us then define the functor G : (Rings) -* (Sets) as follows: G(R) = (i) Set of homomorphisms a : R — R such that rka <_ (n-r) (i. e. , determinants of all minors of a of rk(n-r+l) vanish), together with (ii) Set of homomorphisms b, с Rn 0 R° , (Rm 0 Rn) © (Rn S> R1") @) together with (iii) an R-homomorphism a:R 0R ®R -*R such that the following is commutative 0 Rn (ID and (iv) an R-homomorphism 6 : R 0 R -* R such that (Ш) commute s, and (v) an R-homomorphism • e : R -* R and e : R -*• R such that n e0b-ld л (IV) \/ commutes. It is now easily seen that G(R) can be identified with a subset S*-* R such that there exist polynomials F.(X_,. . . ,X ) over Ж such that l 1 p s = (s^ . . ., s ) « S iff F.(s ,. . ., s ) =0, and the set {F.} and p are
114 115 independent of R. From this it is clear that G(R) is represented by a scheme V of finite type over Ж and since G(R) -* F(R) is surjective, Lemma 5.1 follows immediately. It follows in particular that & is represented by a homomorphism Cp:.SpecP-V (P = A{yy . .'. ,ym]' ). (^) The image of SpecP-{y = 0} in V lies in the smooth locus of V over Spec Z. We have a representation (*> ~+& -* о (homomorphisms of rings and homomorphisms a-s P modules). Now P[z, ...,z] is regular over A and Х-Г is regular over SpecA-{y = 0}. Hence the immersion Xе-* SpecP?z ,. . . , z ] ^T-^- » being an A-morphism, is S P a local complete intersection at every point of Х-Г (we use the fact that a regular local ring which is the quotient of another regular local ring is a complete intersection in the latter; we use this fact for the corresponding local rings of the fibres and then lifting the m-sequence, etc. ). Now codimX in A. is s. Now take a closed point x « SpecP-{y = 0}. Then tensoring P (*) by k(xQ) we get k(xQ) [zr . .., zg] - & 0л k(XQ) - 0 exact. л __ Now Spec & 0Д k(x ) ib precisely the fibre of X over x for the morphism — л — Xе-* Spec P. We claim that Spec С ®л к(х ) is also a local complete inter- intersection in k(x )[z , ...,z ] wherever x € SpecP-{y = 0} (it is a 0-dimensional subscheme of Speck(x ) [z ,.. ., z Iе-* ^L ) and in fact О 1 о К\Х / о that Spec С 0A R*=-* SpecR fz., . .., z "I is a morphism of local complete A p О U 1 SJ intersection over R for any base change Spec R — SpecP-{y = 0} (i. e. , -—; о flat and the fibres of the morphism over Spec R is a local complete intersection). To prove this we note that (T 0Д Rq is locally free (pf rank r) (SpecR -* SpecP-{y = 0}) and SpecR [z ,. . . , z ] -¦ SpecR is a regular morphisra. in particular a Cohen-Macaulay morphism. Now the claim is an immediate con- consequence of Lemma 5. 2. Let B, C, R be local rings such that В, С are R- algebras flat over R and В is Cohen-Macaulay over R. Let f : В -» С be a surjective (local) homomorphism of R-algebras such that С is a complete intersection in B. Then for all R-*R -* 0, the surjective homomorphism о B0R -*C0R -*0 о о (the morphism Spec(C 0 R )c* Spec(B 0 R )) is a morphism of complete inter- intersection over SpecR . Proof. Since the flatness hypothesis is satisfied, it suffices to prove that С 0 к is a complete intersection in В 0 к (к = R/m ), Now we have XV. 0-I-B-C-*0 exact, I = ker В -* С, and I = (f ,. .. , f ), f. is an m-sequence in B. Now the co- dimension of C, Spec(C 0 k) in Spec(B 0 k), is equal to the codimension of Spec С in Spec B, which is s. (This follows by flatness of В, С over R
116 117 and the fact that flat implies equidimensional. ) Let f. denote the canonical images of f. in В 0 к. We have then. (B ® k)/G , . . . ,7 ) = С ® к. Now i Is (B ®4c) is Cohen-Macaulay and the codimension of Spec(C ® k) in Spec(B ® k) is s. It follows by Macaulay1 s theorem that f ,. ¦ . , f is an m-sequence in В ® к. This implies that С ® к is a complete intersection in В ® к, and.proves Lemma 5. 2. The complete intersection trick. We go back to the proof of E). Let X : SpecR — Spec P be a morphism such that R is Artin local and MSpecR )CSpecP-{y = 0}. Consider the morphism (CD » \) : SpecR —V. Then {'CP о \) defirles an R algebra В which is a free R -module of ' о о rank r (in particular flat over R ), and B* is a mbrphism of local complete intersection over R (B is of relative dimension 0 over R ). Let о о R-» R —0 be such that SpecR is an infinitesimal neighborhood of SpecR . о о Then the assertion E) follows if we show that (Q> о X.) : SpecR — V factors 1 о through SpecR-*V. Now В is defined by where a = (a..). Now Spec В -*• SpecR is a morphism of local complete intersection for a suitable imbedding and Spec B. It is clear that Spec В— SpecR is in fact a morphism of global complete intersection for the cor- corresponding imbedding, for it is easily seen that a 0-dimensional closed sub- scheme of A which is a local complete intersection's in fact a global intersection. We have seen in §4, Part I, that the functor of global deforma- deformations of a. complete intersection jljs unobstructed, i_. je. , formally smooth. Hence there is a flat R-algebra 31 such that В1 ® R ^B. Hence th.e mao R ° sequence R —- R — В — 0 can be lifted to an exact sequence R -* R -* B1 — 0; the proof of this is in spirit analogous to imbedding a. deformation (infinitesimal) of Xе-* Ж in the same affine space. A hoirio- morphism (Rn — B) over R is given by n elements in B. Hence this homomorphism can be lifted to R -* B1 and it becomes surjective. Now it is seen easily that Rm -> R can be lifted to R -> R . It follows that о о determinants of all minors of a of rank (n-r+1) vanish. Thus the relations @) above can be lifted to K. Consider the relations (I). We are given b , с such that the following о о diagram is commutative: a ®Id©Id®a_=0_ (Rm 0 Rn) 0 (R*0 Rm) ° With the above lifting of (aj to a we get (Rm 0 Rn) 0 (Rn ® Rn в' (A) -0 ¦ 0. The two rows are exact. We claim that there is a b which lifts b^, and such that the square (A) is commutative. In fact this is quite easy, .for it is clear that we can find V : R ® R -* R such that (A) is commutative. Let b1 be the reduction mod R of b1. Then we see that the composition of о о r
118 119 the canonical map R — В with b -b1 is zero. Hence c о о о b (z)=b!(z) (modlma , or KerRn-*B). о о о о Now b (z)-b' (z) can be lifted to elements in Ima, so that taking for {z } о о a a canonical basis in R -¦ R we define b : R ® R — R b(zQ) = v ea, where 9 с Ьпа lifts b (z)-b' (z). It is now clear that b lifts b and the а о о о square (A) is commutative. This proves the claim. By a similar argument as above there is а с such that с reduces to с and the square (B) is commutative. (Я necessary we can prolong to о the left the exact sequence of the second row in the diagram (*). ) Thus it follows that the relations in I can be lifted to R. /*" A similar argument shows that a , б , e and e which are given о о о о representing the point Spec R -*• V can be lifted to R so that the diagrams (II), (III) and (IV) are still commutative. This means that the Spec Rq — V can be lifted to an R-valued point of V. As we remarked before, the assertion E) now follows. We go back to the usual notations in the theorem. Then: G) _Let CL = m (y , . . . , у ) ideal m A [y] . Then P = А [у] А is_ also the Ql-adic completion of A[y] (of course P is also the m • (A[y"])-adic completion of A [y 1). The proof of this assertion is immediate for a convergent series Zf.,f. € A[y] in the Ot-adic topology is precisely one such that the coefficients i of f, tend to zero in the m -adic topology and the degree of the monomials — oo 1 A This implies that a convergent series is precisely a formal power series in {y.} such that the coefficients (in A) tend to 0 in the m -adic topology, l - ~." A. л . ч: This is the description of P we had and G) improved. л - (8) Now for the morphism Op : SpecP -* V we have that the image of SpecP-{y = 0} is in the smooth locus of V over Ж. We note that V{6b) = (Speck[y])U{y = 0} (k residue field of A), i. e. , V(<Jl) contains {y = 0} so that the image of. SpecP-V(JL) (by Cf ) is also in the smooth, locus л ^ of V over Z. Now the ffL-adic completion of A|_yJ is P. Let P denote. the henselization of (P,tft), P = A[y]. Now apply Theorem 2 proved above. Hence we can find an etale map SpecR-* SpecP which is trivial over V@t) and a morphism CP' : SpecR-* V (note that P is also theOC-adic completion of P ) such that N Cf* = Cp (mod ), for any given N. (Note that the OT.-adic (i. e. , G1R-adic, OL is not an ideal in R) completion of R is also A|y|A). Let &x be the R-algebra defined by y1. Then we have & E&{modOL ) (i. e. , еЧсЬ =&/CTL ). Now Qr* becomes an A-algebra and then we have Ol =& (mod m"^) A N N br (m^R) 2)OO '. Take in particular N = 2. Thus we can find an R-algebra O1 of finite type and consequently of finite type over A such that О-1 =.& (mod Шд). A. Thus to conclude the proof of the theorem, it suffices to prove that (Э*1 is flat over A.
120 (9) Choice of Ort such that a-1 is flat/A. We had a presentation of С as follows: We claim that we have a presentation such that (*) where л I P a . 0 = 0 and РШ ^ РП -> & -* 0 is exact, and 88>k ' a®kA -*0 is exact, where k. = A/m . A A This follows if we prove that Q is flat over A (cf. , §3, Part I). However, (*) can be established directly as follows: We can find an exact sequence of the form A A • P 'kA~' kA - 0. A (Note that Spec С ® kA = X = Spec & is the scheme whose deformation А о о we are considering and that X = ^ ® A/m = Spec & are infinitesimal n A n deformations of X . ) The above exact sequence can be lifted to an exact sequence 121 such that a • 9 = 0, and P -^ Pn — С-* 0 is exact. This proves the existence of (*). Now define a functor G1 which is a modification of G as follows: G! (P0 = Set of {8, a, b, с, а, б, е, с where а, Ъ, с, а, б, е, е are as in the definition of G(R); and 0 is defined by R1 -2 Rm t Rn with a- 8 = 0}. Then as in the case of G(R), we see that G1 is represented by a scheme V1 of finite type over Ж. The given representation for Cf as in (*) above gives rise to a morphism ф : SpecP -*- V . We claim that as in the case of > V, the image of SpecP-{y = 0} lies in the smooth locus of V1 . With the same notations for R, R as in the proof of this statement for the case V, о it suffices to prove the following: Given в е а R'-J>R™_°R*-.B_0 о о о a such that а о 8 = 0 and Rm —2- Rn — В -> 0 is exact, and a flat lifting B1 о о о о over R (hence free over R), we have to lift this sequence to R (the proof of the lifting of the quantities involved is the same as for G(R)).. As we have seen before, for G(R) we have a lifting P ®A/m. A ** * / n+1 n^n . n+1 -~ ®A/mA —* P ®А/тд — & — 0 A An since Cf is Hat over A/m . Passing to the limit, we have an exact П A sequence л/ - Now bn 0 are a set of relations. Since B1 is flat over R these relations о can also be lifted, i. e. , we have a lifting such that а о 9 = 0 and Rm ^ Rn — B1 ¦* 0 is exact. This proves the required
122 123 claim and hence it follows ф (SpecP-{y = 0}) lies in the smooth locus (over Ж) of V1 . Applying Theorem 2, we can find an etale Spec R -¦ Spec P, P = A[yJ and a morphism фх : Spec R -* V such that у = ф (mod OX. ). Let erx be the R-algebra defined by ф\ Then as we have seen before, we have &* = & (mod m*). A. We claim that &x is flat over A. For this we observe that we have a fortiori (mod m ). A. This impHes that &% /m •#' =0/mK& ^ <y . Let А А о (Г) . в' =0, Pm-$>n~ff. ^ exact be a representation of €fx . Recall we have the representation for & (I) л i e ?m a An js, P -* P -* P -* 6" -» 0, and а о 6 = 0. exact Now tensoring (I1 ) with А/т** yields A. л. 6' BA/m" a1 B>A/m° А ^m8A/ ^n®A/"-*a" в A m" - 0, A A and exact (a1 8> A/m*) о (в1 ® A/m^) = 0. We have (I1 ) 0 A/m = (I) ® A/m , as a consequence of the fact that A. A б% = Ъ (mod m ). By (I) I ® A/m is exact. This implies that (I1 ) 0 A/m A. Ai ——i—— . ji is exact. So property, (a1 is exact. So we have that (Г ) ® A/m is exact, and (Г ) ® A/m has the* A • A (91 A. = 0 and ®A/m*- - РП 8> А/тпПл - Cl' 0 A/m11 - 0, A A for all n. This implies, as we saw in the first few lectures,that &x 0 A/m A. is flat over A/m for every n. Now 0X is an A-algebra of finite type and so С1 ® A/m flat over A. A/m for all n implies that ?T! is flat over A (cf. SGA exposes on flatness). A. The proof of the theorem is now complete. Remark 5.1. The fact that &' is flat over A can also be shown in a different manner. This can be done using only the functor G (i. e. , V), but a better approximation (i. e. , better than N = 2) for Ori may be needed. ' This uses the following result of Hironaka: Let В be a complete local ring, b С rn an ideal in В and M a finite В-module locally free (of rank r) outside V(b). Then there is an N such that whenever M1 is a finite B- N module locally free of rank r outside V(b) and M1 ? M (mod b ), then M1 ~ M. Take in our present case В = А [[у]] so that we have R etale over A[y] such that A [y] Id ~ R/Ot and that all the extensions are faithfully flat. Take a coherent sheaf on Spec R; to verify that it is flat over
124 125 A it suffices to verify that its lifting to Spec A [[y]] is flat over A. Take b to be Ob- A[[y]], Then by taking a suitable approximation for &x it follows that the liftings to A[?y]] of &x and & are isomorphic. This implies that & 0 A/m % ffl 0 A/m , hence that O'1 is flat over A, sincel Spec( & 0 A/m11) = X is flat over A. TATA INSTITUTE OF FUNDAMENTAL RESEARCH LECTURES ON MATHEMATICS AND PHYSICS General Editors: K.G. RAMANATHAN and B.V. SREEKANTAN MATHEMATICS 1. On the Riemann Zeta-function. By K. Chandrasekharan* 2. On Analytic Number Theory. By H. Rademacher* 3. On Siegel's Modular Functions. By H. Maass* 4. On Complex Analytic Manifolds. By L. Schwartz* 5. On the Algebraic Theory of Fields. By K.G. Ramanathan* 6. On Sheaf Theory. By C.H. Dowker* 7. On Quadratic Forms. By C. L. Siegel* 8. On Semi-group Theory and its application to Cauchy's problem in Partial Differential Equations. By K. Yosida* 9. On Modular Correspondences. By M. Eichler* 10. On Elliptic Partial Differential Equations9 By J. L. Lions* 11. On Mixed Problems in Partial Differential Equations and Representation of Semi-groups. By L. Schwartz* 12« On Measure Theory and Probability. By H.R. Pitt* 13. On the Theory of Functions of Several Complex Variables. By B. Malgrange* U. On Lie groups and Representations of Locally Compact Groups. By F. Bruhat* 15.. On Mean Periodic Functions. By J.P. Kahane * 16. On Approximation by Polynomials. By J.C. Bur kill* 17. On Meromorphic Functions. By W.K. Hayman* 18. On the Theory of Algebraic Functions of One Variable. By M. Deuring*
126 127 19. On Potential Theory. By M. Brelot 20. On Fibre Bundles and Differential Geometry. By J.L. Koszul* 21. On Topics in the Theory of Infinite Groups. By B.H. Neumann 22. On Topics in Mean Periodic Functions and the Two-Radius Theorem. By J. Delsarte * 23. On Advanced Analytic Number-Theory. By C. L. Siegel * 24. On Stochastic Processes. By K. Ito 25. On Exterior Differential Systems. By M. Kuranishi* 26. On Some Fixed Point Theorems of Functional Analysis. By F.F. Bonsall* 27. On Some Aspects of p-adic Analysis. By F. Bruhat* 28. On Riemann Matrices. By C.L. Siegel 29. On Modular Functions of one complex variable. By H. Maass 30 On Unique Factorization Domains. By P. Samuel* 31. On The Fourteenth Problem of Hilbert. By M. Nagata 32. On Groups of Transformations. By J. L. Koszul 33. On Geodesies in Riemannian Geometry. By M. Berger 34. On Topics in Analysis. By Raghavan Narasimhan 35. On Cauchy Problem. By Sigeru Miz-ohata 36. On Old and New Results on Algebraic Curves. By P. Samuel 37. On Minimal Models and Birational Transformations of Two Dimensional Schemes. By I.R. Shafarevich 38. On Stratification of Complex Analytic Sets. By M. -H, Schwartz 39. On Levi Convexity of Complex Manifolds and Cohomology Vanishing Theorems. By E. Vesentini 40. On An Introduction to Grothendieck's Theory of the Fundamental Group. By J. P. Murre 41. On Topics in Algebraic К-Theory. By Hyman.Bass 42. On The Singularities of the three-body problem. By C.L. Siegel 43. On Polyhedral Topology. By J.R. Stallings 44. On Introduction to Algebraic Topology. By G. de Rham 45. On Quadratic Jordan Algebras. By N. Jacobson 46. On The Theorem of Browder and Novikov and Siebenmann's Thesis. By M.A. Kervaire 47. On Galois Cohomology of Classical Groups. By M. Kneser 48. On Discrete subgroups of Lie groups. By G.D. Mostow 49. On The Finite element method. By Ph. Ciarlet 50. On Disintegration of measures. By L. Schwartz 51. On Introduction to moduli problems and orbit spaces. By P. E. Newstead 52. On Numerical methods for time dependent equations - Application to Fluid flow problems. By P. Lascaux 53. On Optimisation - Theory and Algorithms. By Jean Cea ,54. On Deformations of Singularities. By M. Artin I .55. On Sieve Methods. By Hans E. Richert * Not available