LIBRdRY W S” lllllllfll llllllllllllllllllll llllllllll llllllllllllllllll L 3 1293 10062 8936 THE-3‘3" This is to certify that the thesis entitled On the Homology of Local Cohen-Macaulay Rings presented by Alan B. Evans has been accepted towards fulfillment of the requirements for Ph.D. degree inMiLtLemflliLi fl 7? W Major p0 Date July 25, 1979 0—7639 OVERDUE FINES ARE 25C PER DAY PER ITEM Return to book drop to remove this checkout from your record. ON THE HOMOLOGY OF LOCAL COHEN‘MACAULAY RINGS BY Alan B. Evans A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1979 ,/ Snuwv< ABSTRACT ON THE HOMOLOGY OF LOCAL COHENFMACAULAY RINGS BY Alan B. Evans Let (A,m,k) be a local Noetherian ring. The . l . . . P01ncare series of A is the formal power series 22(2) = 1:0 dimk(Tor?(k,k))zi. A well-known conjecture is that 22(2) is a rational function of 2. In this paper, several formulas giving 2: are derived in the case of A Cohen-Macaulay, that is dimension(A) = depth(A). The embedding dimension of a Cohen-Macaulay ring A is less than or equal to e(A)4—dim(A)-l, where e(A) is the multiplicity. When k equality holds, BA is shown to be rational in Chapter I. A perfect ideal I of A is an almost complete intersection when I is minimally generated by grade(I) +-l elements. In Chapter II, the homology of the Koszul complex of a regular local ring modulo certain almost complete intersections of grade three is computed. From fi’W this, a formula for P is obtained. Furthermore, if I Alan Evans is a perfect ideal of the regular local ring A such that I/I2 has a free direct summand (as A/I—module) of k rank equal to grade(I)-t2, PA/I is computed. In the last chapter, examples are given which show that a composition of Golod homomorphisms need not be a Golod homomorphism and that it does not seem possible to characterize the class of Cohen-Macaulay rings using the deviations, ei(A). Dedicated To Rachael ii ACKNOWLEDGMENTS This paper was written under the guidance of Professor Wei-Eihn Kuan. I would like to thank Professor Kuan for his constant encouragement and Professor R.C. Cowsik for several helpful discussions during its prepara— tion. My special thanks and appreciation go to Mary Reynolds for a superb job of typing the manuscript. iii CHAPTER 0 CHAPTER I CHAPTER II CHAPTER III TABLE OF CONTENTS INTRODUCTION COHEN-MACAULAY RINGS OF MAXIMAL EMBEDDING DIMENSION TWO CHANGE OF RINGS THEOREMS FOR PERFECT IDEALS OF GRADE THREE SOME EXAMPLES BIBLIOGRAPHY iv 18 4O 46 CHAPTER 0 INTRODUCTION Let (A,m,k) be a local (Noetherian) ring with maximal ideal m and residue field k = A/m, and let M be a finite A-module. A projective resolution of M over A is a (perhaps infinite) exact sequence of finite projective A-modules d1+1 d1 (1) ...-’ Pi+l 4 Pia...» Pl 4 PO—oM-vo. See [9],-p.75. M is said to have projective dimension n over A if there is a projective resolution n l (2) O 4 Pn 4 P 4...* P 4 PO 4 M'+ O of length n, but none shorter. The projective dimension of M is infinite if no finite resolution exists. Since A is local, the modules Pi. may be assumed free [33], 3.G. Moreover, k serves as a test module for the pro- jective dimension of M over A (abbreviated hereafter as pdA(M)). Precisely pdA(M) ghn if and only if TorA (M,k) = 0, where Tor?(M,k), the ith left derived n+1 functor of ® [8], p.107, is computed as the ith homology of the complex obtained by tensoring the resolution (1) with k. A free resolution of M d1+1 d1 (3) ...4 Fi+l -9 Fi-4..:’Ed. 4 FO-4D44 O is minimal if di(Fi) E mFi-l for i‘Z 1. In this case, Tor?(M,k) 2 Pi ®A k [33], 18.E. It can be proved that a minimal resolution always exists [33], Ex. 3, p.113. As a consequence, each Tori(M,k) is a finite vector space over k. The ith Betti number of M is defined then to be bi = dimk(Tor?(M,k)). The formal power series PiKz) = Z] bizl is known as the Poincare series of M. i=0 With this notation, pdA(M) < m if and only if 22(2) is a polynomial. It has been conjectured that Pi is in fact a rational function of z for all M. The homological properties of the ring A are connected with problems in geometry. The dimension of A is the length of the largest proper chain of prime ideals m = pn D pn-l D...D pl 3 po [33], p.71. A is called a regular local ring if m can be generated by elements xl,...,xd, where d = dimension of A. Regular local rings correspond roughly to non-singular points on an algebraic variety [34], p.343. For a local ring (A,m,k) of dimen- sion n, the following are equivalent (a) A is regular (b) pdA(k) = n (c) pdA(M)‘g_n for all finite M (d) Pk A is a polynomial (e) 9],:(2) = n+2)“. [33], Theorems 41, 42 and 45. Closest to regular local rings from a homological standpoint are the complete intersections. A sequence of elements xl,...,xr 6 m are a regular sequence if for ). l g.i g_r, xi is not a zero-divisor on A/'(xl,...,xi_l A local complete intersection is a local ring which can be written as a homomorphic image of a regular local ring modulo a regular sequence. The homological characteriza- tion is (a) A a complete intersection if and only if n (b) Pk(z) = Ll4LEl-—' n,m 2 O integers. See [46]. Most progress on the problem of the rationality of P: since Tate's paper [46] has been based on the elegant result of Golod [20]. There, the Poincaré’series is related to homology Operations on H(K) known as Massey products. K denotes the Koszul complex over A, whose underlying graded algebra is the exterior algebra on a minimal generating set for m [33], l8.D. Regular local rings and complete intersections may also be classified using the Koszul complex. Namely, A is regular if and only if Hl(K) = O [24], Theorem 1.4.13 and A is a com— plete intersection if and only if H(K) is the exterior algebra over Hl(K) [24], Theorem 3.5.2. Of frequent use is the fact that P: has a (unique) representation as an infinite product m . s . k _ (l-+221+l) 21 PA(Z) _ .H . e ‘ 0 i=0 (l-221+2) 21+l This was first proved by Assmus using the Hopf algebra structure on TorA(k,k) [2]. The exponents Si = ei(A) are non-negative integers and are called the deviations of A. It turns out that eO(A) = dimk(m/m2), the so— called embedding dimension. Also, if A is a homomorphic image of a regular local ring (R,n), with A 3 R/M and 2 _ . _ . m ELn , then sl(A) - dimk(M/hfl) — dimk(Hl(K)), Where K is the Koszul complex [24], Lemma 1.4.15 and Prop. 3.3.4. The general rationality problem for an A-module M has been reduced to the case M = k, A of dimension zero by Ghione and Gulliksen [19]. In the present paper, three formulas for P: are obtained. The results deal with local rings which are Cohen-Macaulay, a concept which will be explained shortly. Suppose M is a finite A-module. Then xl,...,xr E m form a regular sequence on M if for l g_i g_r, xi is not a zero-divisor on M/(le+u..+uxi_lM). The depth of M, written depthA(M), is the length of the longest regular sequence on M [33], 15.C. The dimension of M is the dimension of the ring A/ann(M), where ann(M) = [x E A[xM = O] is the annihilator of M. In general, the depth of M is at most equal to the dimension of M [33], Theorem 27, and M is defined to be Cohen—Macaulay when depth(M) = dimension(M). A itself is a Cohen-Macaulay ring if it is then-Macaulay as A-module [33], l6.A. The hierarchy of the types of local rings introduced thus far is regular 4 complete intersection = Cohen—Macaulay. The notation used is standard. The minimal number of generators of the finite A-module M will be denoted by H(M). If qig m is an open ideal (mn E q for some n), then e(A,q) = the multiplicity [48], p.294 of q, with e(A) short for e(A,m). Let dA(M) = dimension of M and d(A) = dimension of A itself. For an ideal I‘g m, let ht(I) be the height of I, the infimum of the lengths of saturated chains of prime ideals pn D pn-l 3...: p0, pn a minimal prime containing I [33], p.71. If N is an A-module of finite length [3], p.77, then £(N) = length of N. For an ideal I g_m, the ggagg of I is the length of the longest regular sequence on A which is contained in I. Given any ideal, grade(I) g_ht(I), with equality whenever A is Cohen-Macaulay [18], 11.15. Later, a distinction will be made between the grade of I and depthA(I), the depth of I as A-module. See [18], 21.7° Finally, a system of parameters for A is a sequence x "Xd' d = d(A), of elements from m such that 1).. mn C (Xl’°°"xd) for some n. An alternate characterization of the Cohen-Macaulay property is the requirement that every system of parameters for A form a regular sequence [18], 11.15. When the superscript is omitted, PA will be under- k . . . stood to mean PA, and Will be called the POincaré'series of A. CHAPTER I COHEN-MACAULAY RINGS OF MAXIMAL EMBEDDING DIMENSION Let (A,m,k) be a local, Cohen-Macaulay ring. Under the assumption that k is infinite, Abhyankar showed that eO(A) g_e(A)4-d(A)-—1 [1]. This restriction on k is not important for the study of the Poincare series. it Define A = A[X] the localization of the polynomial m[X]' ring over A in one variable at the prime ideal m[X] = [f = ZDaiXilai E m]. By passing from A to A*, the residue field may be assumed infinite [38], Ch. IV. From [24], Prop. 1.9.8 and Lemma 3.1.2, it follows that ei(A*) = ei(A) for i 2 1, and eO(A*) = €O(A), by a result of Lech [29], Lemma 2, p.75. Thus from the infinite product representation, PA = PA*. NOW, eO(A) = e(A)4—d(A)-—l if and only if - 2 _ (x1....,xd)m — m for some x1....,xd E m, d - d(A) [41], Theorem 1. In the case d = l, m is said to be stable [31], and the rationality of the Poincare series has been established in [13]. The following is an ex- tension of this result to higher dimensions. Theorem 1.1. Let (A,m,k) be a local Cohen— Macaulay ring of dimension d 2_1 with eO(A) = e(A)4— d(A)-l. Then (1+2)d PA(Z) = l-(eO(A)-d)z ° Proof: As mentioned earlier, the case of interest . 2 . ' is d‘z 2. Assume (x1....,xd)m = m . First, note that x1....,x form a system of parameters for A, because d .Vflxl,...,xd)m.=.¢Qxl,...,xd) fl.¢fi =‘/(X1’°°"Xd) =¢4§= m [3], p.9. Next, at least one of the xi must lie in m‘xmz. Otherwise, m3 2_(x1,...,xd)m = m2, contradicting Krull's intersection theorem [33], Cor. 2, p.69. Now, for l g_i g_d-l, the dimension of A/le,..., ) is d-i x. 1 [33], 12.K. Moreover, (§i+l,...,§d)a = (5)2, where the bar denotes the residue class in A/(xl,...,xi). Because x1....,xd form a regular sequence [18], 11.15, A/le,...,xi) is again Cohen-Macaulay [33], p.104. By induction on d, §i+l lies in m \(m)2, hence x. 6 mfl\m2. Therefore all of x ,...,x lie outside 1+1 1 d 2 m 0 Let A =.A/(X2....,Xd). Then (il)fi = (5)2, so that m is stable. Because X1”°"Xd form a regular d‘1 p (z) [24], Cor. 3.4.2. But since m is stable, P_(Z) lCFE A l-(eO(A)-l)2 sequence in m‘\m2, PA(z) = (l+—z) H WI [13]. Furthermore, reducing modulo each xi decreases the embedding dimension by one. To see this, note that xl,...,xd can be extended to a minimal generating set d n for m. Suppose that Z} c.x. = Z) y.z. 6 m2. Since ._ i i ._ j i i-l 3—1 2 d (xl,....xd)m = m , .Z) cixi = .Z) wiXi With wi E m. i=1 i=1 If some c. fl'm, then from x.(c.-W.) = Z} w.x. it i i i i j#i j 3 follows that c x (l-c_lw ) = Z) W’X and so i i i i . . ' " #1 -l -l -l . x. = c. (l-c. w.) Z} w.x. , i i i . . l 3311 3 3 contradicting the fact that X1"”’Xd form a regular sequence. Thus ci 6 m for i = 1,...,d and x1....,xd remain linear independent in m/mz. Then for . 2 . i — 1,...,d, xi E'm modulo (Xl""'xi-l)' NOW uSing [18], 1.32 d-l times, 30(A) = eO(A)-(d-l). Hence (1+2)d PA(Z) = 1- (60(A) -d)z ' as claimed. Remark 1.2.1. A is Cohen—Macaulay if and only if * A is Cohen—Macaulay [11], Theorem 2. Moreover, the dimension and multiplicity are invariant under the * passage from A to A [32], Prop. p.277. As noted lO * * earlier, eO(A) = eO(A ), so A has maximal embedding dimension Whenever A does. Remark 1.2.2. If eO(A) = e(A)4—d(A)-—l, with eO(A)-d(A) 2.2, then A is not a complete intersection, as is easily seen by comparing P with the Poincare series A (l-QP (1-22)m of a complete intersection. Before proceeding, a brief review of projective varieties is in order. Let k be an algebraically closed field of characteristic zero. Projective m-space over k, written 113:1, is defined to be the set of all points (XO’X1""'Xm) # (0,0,...,O), xi 6 k, modulo the equiva- lence relation (x0,xl,...,xm) ~ (1x0,1xl,...,kxm). A E k \[O]. A projective variety in E31 is the locus k of zeros of a finite set of homogeneous polynomials f fr 6 S = k[XO,Xl,...,Xm] such that f f 1’°°" 1""’ r generate a prime ideal in 8. Associated to each projective variety X E 113;: is its homogeneous coordinate ring, S/I(X), where I(X) consists of all forms in S which vanish identically on X. An ideal I of S is graded if I can be generated by forms. Then projective sub- varieties of X correspond to graded prime ideals containing I(X), with the exception of the irrelevant maximal ideal (X0,Xl,...,Xm), which contains every graded ideal. If one ignores the grading and considers the locus of zeros m+1 of I(X) in affine m4—l space Jfi< , one has the cone associated to X, C(X). C(X) is an affine variety 11 through the origin in..A§?l’ which has dimension one greater than the dimension of X. Furthermore, many geometric pr0perties of the vertex of C(X) are closely m k 0 linked to the geometry of X back in 3’ Conversely, if R is a graded k—algebra of finite type, there is a geometric object Proj(R) associated with R [25], p.76. In case R is the homogeneous coordinate domain of a projective variety X, then Proj(R) a X. Let R = 6 R. be the homogeneous coordinate domain iZQ of a variety X of dimension d contained in IPE. Then there is a polynomial P of degree d ‘With rational X coefficients called the Hilbert polynomial of X such that Px(n) = dimk(Rn) for sufficiently large n. The degree of X, written deg(X), is d! times the leading coefficient of Px(n) [35], 6.25. The integer deg(X) tells the number of points in which "most" linear subspaces L _<_:_ IPI}:t of dimension m-d meet X [35] , Theorem 5.1. At the lower end, pa(X) = (—1)d(PX(O)-1) is called the arithmetic genus of X and is an important geometric invariant [35], p.115. Now, in order to produce a Cohen-Macaulay ring of maximal embedding dimension, the following fact will be useful. 12 Lemma 1.3. Let X g 11311:I be a projective variety of dimension d over an algebraically closed field k of characteristic zero. Let (A,m,k) be the local ring at the vertex of the cone C(X). That is, A is the localiza- tion of the coordinate ring of X at its maximal ideal (XO,X X ). Then e(A) = deg(X). l'OOO'm Proof: Let R be the homogeneous coordinate ring of X. Let Px(n) be the Hilbert polynomial of X and let PA(n) be the characteristic polynomial of A [48], Ch. VIII. Consider Grm(A) = o ml/ml+l, the associated iZQ graded ring of A [33], lO.C. Since A is already graded, Grm(A) a R, or geometrically, Proj(Grm(A)) a X. There- fore, Px (n)— -dimk(m n/mn+l) for large n. By definition, xn A(n) = 1230 dimk( m i/'mi+l) for large n [48], Ch. VIII, Theorem 19. Therefore, Px(n4-l) = PA(n4-1)-PA(n) for sufficiently large n. But PA(n4-l)-P A(n) = P1;(n)+ Ql(n) . Where PA is the formal derivative of PA with respect to n and Ql(n) is a polynomial of degree less than deg(PA)-l. This means that PA(n) is the indefinite inte- gral of Px(n), plus a polynomial Q(n) such that de (Q) de (P ) NOW P (n) - Eiélflf:i lus ter f g < 9 A ' A - (di-l)! p ms 0 lower degree [48], Ch. VIII, §10, and Px(n) = E—gggiél plus terms of lower degree [35], 6.25. Therefore, §%%H§§j1-=j‘E—Q§$L—L, the indefinite integral with respect nd+l to n, which of course equals n(d4?i?§X) . Hence e(A) = deg(X). 13 Example 1.4. Let Y be the curve in P2 defined by the equation XOXl = X3. IP2 is covered by affine Open sets D0 = {x0 7! 0}, D1 = [x1 '71 o}, [)2 = {x2 7; o) [35], 2A, on which the equations of Y are X X X X X 1 2 2 O l . X7. , 27- (if) , and (if)(§f0 = 1, respectively. 0 O l l 2 2 Therefore, Y is smooth by the Jacobian criterion [25], p.31. Consider the Segre’ embedding X = S(Yx 1P1) g S(IP2x I91) g P5 (see [35], 2B). Since Y is rational, [44], p.6, and since the arithmetic genus is a birational invariant [25], III, Ex. 5.3, pa(Y) = pa(fl?l) = O. [25], I, Ex. 7.2. So a... Hilbert polynomials are PY(n) = 2n+ l, PE>1 (n) f n+-l. By a theorem of Seidenberg, Px(n) = PY(n)‘-PEA_(n) = 2n24-3n4-l [42], Theorem 2. Thus deg(X) = 4 [35], 6.25. Furthermore, X is arithmetically Cohen- Macaulay [45], Cor. on p.374. That is, (A,m), the local ring at the vertex of the cone on X, is Cohen—Macaulay. A is then a Cohen-Macaulay ring of dimension three and embedding dimension six whose multiplicity is four, by Lemma 1.4. Therefore PA(z) =-;%%}§%— , by Theorem 1.1. A does not seem to fit any previously known criteria for rationality. Suppose (A,m,k) is a local ring. Then there is a natural ring homomorphism from Z, the ring of integers, to A which sends l to l. The generator of the kernel of this map is called the characteristic of A. A is said to be equicharacteristic if char(A) = char(k). Let A be 14 an equicharacteristic zero, Cohen-Macaulay local ring of dimension one. Two conditions which are known to imply the stability of m are (l) A saturated [31], Cor. 5.3 and (2) A seminormal [12], Theorem 1. For higher dimensions, neither imply that A has maximal embedding dimension. In fact, the stronger hypothesis of normality does not suffice. Egample 1.5. Let k be a field, for simplicity algebraically closed of characteristic zero. Let [Xij], i = 1,...,s; j = l,...,r with s < r, be indeterminants and let R = k[[Xij]]. Define I to be the ideal generated by the s>37 V: {Xij}: l<(n-l) alternating matrix obtained from g by deleting the ith row and ith column. In addition, P admits a commutative, 20 associative algebra structure compatible with d [5], 8.4. Let Li' i = l,...,n, Mi' i = 1,...,n and N be bases for Pl,P2 and P3 respectively. The multiplication n ' . . = .. P .. , .M. = . . = .. , on P is LlLJ “£21 013k fijk(g)Mk L:L ] MJLl éle Where for i,j,k 6 [l,...,n], Gijk denotes the sign of the permutation (i,j,k,[l,...,n}‘\[i,j,k]) and Pfijk is obtained by deleting rows and columns i,j,k from 9. Let Klzo 4 K ‘4 K -4 K. 4 A 4 A/x 4 0 be the Koszul 3 2 1 resolution of .Aflx, and write dK for the differential on K, dP for the differential on P. Then K1 2 ATl @ AT2 @ AT3, 1 With dK(Ti) = Xi' Since §.E I, there eXist X1,X2,X3 6 Pl such that dP(Xi) = Xi' Defining wl(Ti) = Xi and extending by ¢2(Tqu) = Xqu, ¢3(T1T2T3) = XlX2X3, gives a map w:iK-4 P which is a homomorphism of differential graded algebras: P: 0-4 P 4 P 4 P 4 P 4 A/U 4 0 ($3 ((2 ’11 ¢= K:O4K4K4K4K4A/I4O The construction of w is sketched on page 472 of [7]. Now diN = 0 implies the relations n i+l (201) Z; (-1) y-z'u = O, for j: l,...,n. i= 1 ij 1 21 n Let Xi = E: biij° Computing. one has n n (2.2) prq = (jig bijj)(#E£ bquk) n n n = jig PE: bpjbqkpggi ijflpfjk£(g)ML] n n n Z] Z) Z) j=l k=l i=1 b f ijz pjbqu jk£(g)ML’ Because K is an exterior algebra, T: = O, p = 1,2,3, which implies X2 = 0, since ¢ is an algebra homomor- phism. So, n n n Z3 Z3 b .b Z) c. Pf. M = o j=l k=l pj pk[k=l jkL jk£(g) L] which implies n n 2.3 Z3 Z3 b .b o. Pf. = o, for z = l,...,n ‘ ) jzl kzl p3 pk :kz Jk1(g) and p = 1,2,3. Also, (2.4) ¢3(T1T2T3) = XlX2X3 n n n = ( Z: b .L.)( Z} b )( Z) b L ) j=1 13 3 k=1 2kLk i=1 32 2 n n n = ( Z3 Z) Z) b .b b o. Pf. (g))N. j=l k=1 £=1 1] 2k BE jkL ij There are relations determined by the fact that w is a chain map. First, dP 0 $1 = Id 0 dK implies (.1)3+1 b. y. = x. for i = 1.2.3. 2.5 < > 1 13 j 1 "[45 3 Since dP 0 $2 = $1 0 dK, dP(Xqu) = ¢l(xqu-qup), so 22 n n b L - b L n n n d ( Z} Z) Z) c. b .b Pf. (g)M ) P j=l k=l £=l jkfi pj qk jkfl E n n n n = Z V Z O. b -b Pf. .L. 0 i=1 ktl L=l 3k“ P3 qk 3k‘(g)(F§1 2‘1 1) Therefore, (2.6) x b .-x b n = Z) Z Z G. b .b Pf. ., j=1 k=l i=1 jk£ pj qk jk£(g)z£i ._..-llHll|||| . Finally, 12 ° dK(TlT2T3) = q’2‘X1T2T3""‘2Tir‘r3""‘3TiT2) n n n = x ( Z) Z) Z) c. b .b Pf. (g)M ) 1 j=l k=1 2:1 gkz 23 3k jkL 2 n n n ' X2( 4:3 Z" Z ijzbljb3kpfjkz(g)Mz) n + x (.23 Z} Z} ijzblijkajk£(g)ML) n n n _ T‘ ‘ dP(an k:& 2;: ijzbljbzkb3zpfjkz(g)N) = -1 .M. . . . Therefore, I1 n (2.7) x ( Z) Z) 0. .b .b Pf. .(g)) l j=l k=l jki 2] 3k jki n n (2 Z 0 .b .b Pf.ki(g)) _. x . j=l k=l jki 1] 3k 3 2 n 1'1 + x ( Z} Z) o. .b .b Pf. . 3 j=l k=l jkl 13 2k jkl(g)) 23 n . n 1'1 1+1 (-1) “(32:1 kill 3:31 Ojkzblijkb3£Pfjk£(g)) NOW, Peskine and Szpiro have shown that V V V V . F : O 4 Pl 4 (KlOPZ) 4 (KZOPB) 4 K3 4 A/I 4 O is an A-free resolution of A/I [40], Prop. 2.6. The notation ( )V is that of Peskine and Szpiro for the A-module dual. One obtains F by reducing Q, the mapping cone of ¢:ZK-4 P, modulo the subcomplex KO O O 4 O o P and then dualizing. 0 Let 0 stand for the reduced mapping cone. Recall that the mapping cone of a map O :U’4 V of complexes is the complex ...4 Ui O Vi+l 4... endowed with the differential 4 d(a.b) == (—dU(a). Cp(a)+dv(b)). Note that F =_-_ A, F e: A , O 1 n+3 n ' ~ F2 2 A , F3 2 A . Furthermore, according to Prop. 1.3 of [7], F possesses the structure of a commutative, associative, differential algebra. What follows is an explicit calculation of a multiplication table for F. As in [7], define 82(F) = (F69F)/M, Where M is the graded submodule of F ® F generated by {f®g - (-1) (deg f) (deg g) g ® f|f,g 6 F both homogeneous}. 52(F) is a complex with s (F) a-.( Z F.®F.)+G , where 2 k i+j=k i j k iT3 “.2 (-1)i+l y.M.) )T 1:1 i i + (aix3"a3xi 2' n n n + a ( Z} Z) Z} 0. b .b Pf. (g)M ) 3 i=1 j=l k=l jkfl 1] 2k jkL L n n n Z) Z) Z) 0. b .b Pf. (g)M 2:1 j=l k=l 3k2 13 3k 3k2 ) +a 2( 2 n n n (2 23 Z30 b .b Pf (g)M£>) by (2.2) + a . . 1 2:1 j=1 k=1 jk2 2] 3k jk2 which equals . n n 1+1 (-1) by.-+a ( Z3 Z) c. .b .b Pf. .(g)) i 3 j=l k=l jki 13 2k jki 2 <3 +a( 0. .b .b Pf. .(g)) 2 j=l k=l jki 13 3k jki <2 2 -+a 0. .b .b Pf. . . l(j=l k=l jki 2] 3k jkl(g)) So, . n n i+1 = _ 0 (2.8.3) dci ( l) yiA4't(,§3 §> jkiblijkajki(g))Al j—l k-l n n +(Z Z] c. .b .b Pf. .(g))A j=l k=1 jki 13 3k jki 2 n n +(Z 230 b.be j=1 k=l jki 23 3k jki(g))A3’ 3 n Finally, let (a,c) = (kg: aiTi’ fig: ciMi) 6 K1 ® P2. Then (dDi)(a,c) = DidQ(a,c) = Di(—dKa,ch+-¢l(a)) 2 2 2 = D (O, c z .L.4— a ) l k=l 3:1 k k3 3 k=1 kxk (since Kl = O in Q) =D.(O, _ Z c 2 .L. + ‘ b .L.) l k=1 3:1 k k3 3 k=l 3:1 k3 3 2 2 2 = D.(O, ( c z .-+ ‘ a b .)L ) i 3:1 =1 k kj :1 k kj n 2 = Z} c z .-+ b .. =1 k k k=l ak k1 Therefore, 3 n (2.8.4) dD. = Z: b..B.—t Z) z..C . l j=l 31 j=1 )1 3 To compute the products in F, notice first that 21(f®l) = f, 21(189) = 9. Thus @ld(Al®A2) =- §l(dAl®A2-Al®dA2) = -x3A2-x2Al, @ld(A2®A3) = x2A3+xlA2, and @ld(Al®A3) = -x3A3+xlAl. Hence §2(A1®A2) = A1 0A2 = -Bl 22(A1®A3) = A1'A3 = '32 §2(A2®A3) = A2 .A3 = -B3. Clearly Ai oAi = O for i = 1,2,3. n Next, d(- Z} b .C.) by (2.8.3) equals ._ 3i i i—l “ 2 2 2 (_ Z) b .y.)A -( b . o. .b .b Pf. .(g))A i=1 3i i 4 i=1 31 j=l k=l jki 13 2k jki l n n n -( E b . Z) Z} b .b Pf 31 O3ki 13 3k jki(g))A2 i=1 3:1 k=1 28 n n n -(2 b, 2 z: o. .b.b Pf. .(g))A i=1 3i j=l k=l jki 23 3k jki 3 n n E) ~ =-xA-(Z Z3 b.bb Pf. (g))A 3 4 j=l k=l 2:1 1] 2k 32 jk2 l 2 2 n - ( b .( Z) c. .b b .Pf. .(g))A j=1 1] k=1 i=1 jki 3k 31 jki 2 n n n ._( 2) b ,( Z) Z) o. .b b .Pf. .(g))A j=l 23 k=l i=1 jki 3k 3i jki 3 n n n = -xA —(Z} Z 23 b .b b Pf. .(g))A (use (2.3)) 3 4 j=l k=l i=1 1] 2k 32 jki 1 = 61(d(Al®A4)) by (2.8.1). n 1 Therefore Al A4 - -‘E) bBiCi° Similarly, A2 ~A4 = i-l n n _ o = — b ° = E) b2]..ci and A3 A4 Z licl. Clearly A4 A4 0. i—l l—l Progressing to products of higher degree, §2d(Al® Bl) §2d(Al®B2) §2d(Al® B 3) §2(dAl® Bl) - 42(Al®dBl) _-. §2(-X3Bl) - 92(Alo (x2A1+x3A2)) = —X3Bl—X2Al 'Al+X3Bl = O. = -x B --2 32 2(A 1 ® (x3A3 - xlAl)) = —x382--x3Al -A34-xlAl oAl = -x B 4-x B = 32 32 0' A deB =<22(‘m18’13’3)‘§2( l 3) = -x3B3 - 22 (Al 8 (—xlA2 — x2A3)) = “X333 " X1131 ‘ X2B2° 29 n . 3 n 1)j+1yb.)—-Z (1)3+1 (2: b 3+2: > 3 3 j=l 1:1 13 1 i=1 13 l 3 n = Zj+ly ybl)B i=1 j= —1 3 n n . + Zc i—l j=l J l] l i=1 = 22d(Al®B3) n i+l Therefore, A1 B3 = - Z} (—1) YiD Similar calculations i=1 yield A2 . l = A2 ~B3 = A3 -82 = A3 ~83 = O and _ i+l _ _ . AZ-BZ— (-1) le — A3 B1. Next a d(A4®B ) - 32(dA4OBl-A4Odsl) n =§2((Z 23 beb Pf-(g))B) j- -1 k-l 2-1 lj 2k 32 ijL 3k2 1 + 22"((X2A1X3A2)®A4) n n n =(23 Z be b o. Pf. (g))B j=1 k=1 2:1 13 2k 32 jkE jkz n n = (:3 E E blijkb320 3k2P fjk2(g))B n ‘ Z (b3ix2‘x3bzi)ci n n = ( E Z 23 b1310219932C7 3k2 Pfjk2(g))Bl n - Z) Z} Z} Z) c b b Pf . 2. C 2:1(j=1 k=l i=1 jki Zj 3k jkl(g) 12) by (2.6). 30 n “.2 Z Z ijzijkaPfjk2(g)DL) n = 3E1 #23 2E: Ojk2b23b3kpf3k2(g)(bl2Bll'b22B21'b32B3) n n = ijinjb3kajki(g)zi2)Ci 2:1 j=l k=1 1 l b b .b Pf (g))B O3k2 12 23 3k jk2 1 n 221 Ojk2b22b23Pf3k2(g))Bz Cjk2b3kb32Pfjk2(g))B3 l n - Z3( Z3 Z3 Z3 0 .b .b Pf. 1:1 j=l k=l i=1 jkl 2] 3k Jki(g)zi£)ci = §2d(A4®Bl) by (2.3). Therefore n n n a (A 233 ) = A °B = Z3 Z3 Z) c. b .b Pf. (g)D . 3 4 l 4 1 j=l k=l 2:1 jkl 23 3k jkfi 2 Similarly, n n n A .B = Z3 Z3 Z3 0. b .b Pf- (g)D 4 2 3:1 k=l 2:1 jkz 13 3k jk2 L and A .B = ‘ Z) c b .b Pf (g)D . 4 3 3:1 k=l 2:1 k2 13 2k k2 2 Continuing, §2d(Al®Ci) = C22(dA18Ci) -22(Al®ci) . n _ 1+1 — x3C -+(-1) l( E) bBECL) 31 I1 II + ( Z3 Z) c .b b Pf .(g))B j=l k=1 jki lj 3k jki 1 + (jg: £51 Gjki b23b3kpf3 ki(g))B2 since A °A = 0. However, n n d(Z) Z) c. .b Pf. .(g)D.) j=1 k=l jki 3k jki 3 I1 n 11 I1 + ( Z) Z) O b b Pfjk i2(g))B j=l k=l jki 2j 3k I1 I1 I1 + ( z) z) o. .b Pf. .(g))( Z) z .c ) by (2.3). j=l k=1 jki 3k jki 2 1 2] 2 Also, n n 2% Z) G. .b Pf. .(g)z .C 2=l j=1 k=1 jki 3k jki 23 2 E5 n = Z3 b ( Z3 6 .-(g)2 -)C 2=l k=1 3k j=l kij 23 2 since F (-1)1 yi if 2 = k n ( 3E1 Gki lj Pfkij(g)ztj = o if 2 4 k K n k+1 and Z) G’ Pf (g)zi . = (-l) y , by the formula on j_ —1 k11j ki1j13 k page 443 of [5]. So by (2.5), x3Ci+(-l)l 5751,2110 ) 32C 2 I1 I1 n = 2 E Z ijib3kajki(g)z2jc2 2:1 3 l k=1 I1 I1 and thus, d( E :3 OjkikaPfjki(g)Dj) = @deloci). j=1 1 32 Therefore, A1°Ci = §3(A1®Ci) = g: g Ojkib3kpfjki(g)Dj' j—l k~l Similarly, I1 I]. A2 "31 = §3(A2®Ci) = :43 :43 Ojkinkajki(g)Dj j—l k-l and n n A3 .ci = §3(A3®C_i) = 321 131 Ojkiblkajki(g)Dj. I have been unable to find an explicit formula for products of the form A4 .Ci' This turns out to be unnecessary for the proof of Theorem 2.11 below, since it will be shown that the corresponding products vanish in F ® k. Putting together the above results gives Table 2.9 Al°A2=-Bl Al Al=A2 A2=A3 A3=O A .213=-132 A4-A4=O A2°A3‘1’B3 n n n A1°A4 = ‘2 b3lci A2 A4 = '1? b2 1 A3°A4='Z blicl l=l 1—1 1:1 Al-B1=Al A2=O A2.Bl=A2.B3-—-o n . I1 +l . i+l A-B=-23(1)l D AoB=Z(1) l 3 i=1 1 l 2 2 i=1 i l A3-B2=A3-B3=O I1 . 1+1 A3 B1 — 523 (-1) 1D; 33 Table 2.9 continued n n A ~13 = 2 Z E o. b .b Pf. (g)D 4 l j=l k=l 2:1 jk2 2] 3k jk2 2 § 2“: n A °B = Z) G. b .b Pf. (g)D 4 2 j=l k=l 2=l jk2 13 3k jk2 2 n E} n A -B = '23 2‘; o. b .b Pf. (g)D 4 3 j=l k=l 1:1 jk2 13 2k jk2 2 <2 i A °C. = G. .b Pf. .(g)D. l l j=l k=l jkl 3k jki 3 2‘5 § A ’C. = G. .b Pf. .(g)D. 2 l j=l k=l jki 2k jki j n n A cc. = 23 23 o. .b Pf. .(g)D.. 3 l j=l k=l jki lk jki 3 Before going farther, some terminology from the homological theory of local rings must be introduced. The general reference is the book of Gulliksen and Levin [24]. Let (R,m,k) be a local ring and let X be a differential graded algebra whose graded pieces are R-modules. X is strictly skew-commutative if xy = (-l)pq yx for x E Xp, y e xq and x2 = o if deg x is odd. x is a divided power algebra if to every element x E X of even positive degree there is a sequence of elements X(L) 6 X, 2 = O,l,2,... satisfying (1) X(O) = l, X(l) = x, deg x(£) = 2 odeg x (2) X(j)X(£) = ”ii/”((342), where (j'£) =_£.j_j+_‘£_.:_:. i+j=2 fl 34 (4) for 2.2 2 0 if deg x and deg y are odd (2) (2) _ (XY) "' 2 x y if deg x is even and deg y is even and positive (5) (x(j))(£) = [j:2]x(j£) for 2.2 O. j.2 l, '2 1 where [j,2] = fl 21(j1) An (R)—algebra is a strictly skew-commutative, differential graded algebra endowed with a system of divided powers Which is compatible with the differential. An (R)—algebra X is assumed connected, that is l -R = X where l is the ol identity of X, and will usually be augmented over k. If f':X + Y is an (R)-algebra homomorphism, let FqY be the (R)-subalgebra of Y generated by f(X) and all the elements of Y of degree less than or equal to q. {FqY}qZO is called the filtration associated with f. A homomorphism f :X«+ Y is said to be a free extension when Y'a X ® ( ® R ), Where R is the (R)—algebra iEI 1 formed by adjoining the variable Si to the trivial (R)— algebra in the following fashion. If deg Si is odd, then R denotes the exterior algebra generated by Si. If deg Si is even, R is the polynomial algebra in a countable number of variables Si = Sil), 8:2),... with relations 5:3)séfi) = ilj§%%i-S(j+£) for jI2 2,1. * Let X be an augmented (R)-algebra. The pair (X ,h) is an acyclic closure of X if 35 'k (l) h.:X + X is a free acyclic extension and (2) 1308") g C(x*) +mx*+x. * * Here C(X ) denotes the decomposable elements of X . That * is, C(X ) is the submodule generated by all elements xx’, where x and x’ have positive degree, together with all divided powers y(£) , 2 > 1, y of even positive degree. It turns out that a free acyclic extension X‘+ Y of the canonical augmented (R)-algebra X':Ru4 R/m is an acyclic closure of X if and only if Y is a minimal resolution of k. Gulliksen has proven the existence of the acyclic closure and in particular, the existence of a minimal (R)—algebra resolution of k [22]. Returning to almost complete intersections, by making an additional assumption, it is possible to determine the homology of the Koszul complex with the help of Table 2.9. First, Lemma 2.10. Let (A,m,k) be a regular local ring and let I Eimz be a (perfect) almost complete intersection of grade three. Suppose further that I = (xl,x2,x3) :J, J = (yl,...,yn) Gorenstein of grade three as above, and n . _ _ 3+1 ~ A x. — E: ( l) bijyj' With bij E m. Then c3(A/I)‘2 n+-3. Proof: The resolution F * A/I is minimal precisely when bij 6 m for all i,j. Thus TorA(A/I,k) E F 8A k. Call this algebra A. Then A inherits the following algebra structure from F :Al °A2 = -Bl, Al -A3 = -B2, 36 A2 0A3 = -B3, with all other products zero, except perhaps those of the form A4 °Ci. Let K = KLA/I) a K(A) ®A (A/I) be the Koszul complex over A/I. Then there is an iso- morphism. m :A » H(K) of differential graded algebras which preserves Massey products [5], p.401. First of all, dim H1(K) = dim A1 = eltA/I) = u(I) = 4. Let X a k be the augmented (A/I)-algebra given by K, and let h.:X * X* be the acyclic closure of X, that is, a minimal (A/I)- algebra resolution of k. Let {FqX*} be the filtration associated with h. Define 21,22,23,z4 E Zl(X) to be cycles of degree one such that 2i = w(Ai), i = l,2,3,4, where the bar indicates the residue class in H(K). Adjoin variables 51,82,8354 of degree two to X With dSi = zi, * i = l,2,3,4, so that X E.F X . Then d(ziS4) = 3 (dzi)S4--zidS4 = -ziz4 = -51 = ~dli, i = 1,2,3. Define V. = -1 -z.S l l l 4, i = 1,2,3. One has dVi = -6i-d(ziS4) = O, * so Vl,V2 and V3 are cycles of degree three in F X . 1,92 and v3 are linearly independent in H3(F * .. 3(F3X ), the reduced homology, because 2 3 * 3X)‘ 1’22 and 23 are themselves linearly independent in H(K). Moreover, \7 E V1.92,93 and m(Di), i = l,...,n are linearly independent because m(Di) 6 X for i = l,...,n and S4 is an indeterminant over X. Combining the homology classes Vi ~ * with cp(Di) gives dim(H3(F3X )®k) 2 n+ 3, and by [24], ~ * Lemma 3.1.2, e3(A/I) = dim(H3(F x )®k). 3 Theorem 2.ll. Suppose (A,m,k) is a regular local ring, and let 1'; m2 be a grade three ACI with I = xin) 37 J = (yl,...,yn) Gorenstein of grade three, where n 3+1 xi = Z) (-l) bijyj' bij E m, as in Lemma 2.10. Then j=l Hl(K)i-H2(K) = O, and there eXists a baSis 21,22,23,z4 . 2 _ - for H1(K) such that dim(Hl(K) ) _ 3 and 21 .24 _ z oz4=23oz4=0. Proof: Most of the calculations were done above. 2 1 . 2 . three Since Al 1.82 and B3. It remains to ShOW’that H1(K)i-H2(K) = O. From a formula due to The dimension of Hl(K)2 is the dimension of A which is is spanned by B Levin, Sakuma and Okuyama [24], Theorem 3.3.4, 6 63(A/I) = dim/H1(K> -H2(K» +(21) - dim2> which is greater than or equal to ni—3, by Lemma 2.10. But dim(Hl(K)2) = dim(Ai) = 3, 61 = 4 and dim H3(K) = dim A3 = n, so n+ 3 g n-dim(Hl(K) ° H2(K)) + 3. That is, H Remark. The purpose in determining the structure of H(K) is to be able to compute the Poincaré series of A/I. Since PA/I is determined by H(K) [4], Cor. 5.10, and since the structure of H(K) obtained above is the same as that computed for almost complete intersections of embedding dimension three by Golod in Proposition 1 of [21], it would follow that 38 Pk (z) = (1+2)d A/I l-z-322+(3-n)23-25 But this result could be obtained without knowing H(K), by first reducing modulo a regular sequence of length d-3 in m‘xmz. NOW the determination of H(K) is of inde- pendent interest. However, there exists a method of reduction to dimension zero which avoids the somewhat complicated computations above. Pick a regular sequence x1,...,xd_3 of length d-3 in m‘xm2 which remains regular under reduction modulo I. Upon reducing modulo xl,...,xd_3, the homology of the Koszul complex is invariant [6], Prop. 1. So it is not surprising that the structure determined in Theorem 2.11 coincides with that in [21]. Consider an arbitrary ideal I gim of the regular local ring (A,m,k). As is well—known, .A/I is a complete inter- section if and only if the conormal module I/I2 is free as .A/I—module [l6],[47]. As a final note in this Chapter, it is shown that local rings which are close to being com— plete intersections in the above sense have rational Poincare series. Theorem 2.12. Let (A,m,k) be a regular local ring of dimension d, 1.; m2 perfect of height r which is not 2 a complete intersection, such that I/I has a free direct summand of rank r4-2. Then PX/I is rational. 39 Proof: Suppose I/I2 a (A/I)r"2 @ N. This means that there exist X1”'°’Xr-2 E I and an ideal J, I 3 J': 12 with (xl,.. 2 E_I . In fact, x1....,xr .,xr+2)4-J = I, (xl,...,x H J r-2) _2 can be chosen to be a * regular sequence [47], Lemma 3. Let A = A/(xl,...,xr_2), 'k I = I/(xl,...,xr_2). Then I/xlI, and hence I/xlA, has finite projective dimension over A/xlA, [47], Prop. 1.2 and Lemma 2. By the "triangle inequality", [18], * * 18.3, I has finite projective dimension over A , as remarked in [47]. Then by a change of rings theorem, [18], * 18.7, since pd *(I ), and hence pd *(A/I), is finite, A A pd *[A/I) = pdA(A/I)-(r-2) = 2, as I is perfect. A * Furthermore, grade *(I ) = gradeA(I)-(r-2) = 2 [24], A * * Prop. 1.4.7, which equals ht *(I ). Therefore, I is A * perfect of height two in A . As a consequence, * pd *(I ) = l [18], 18.1, and it is known then that A k P k(z) = PA*(z) A/I l-nzz—(n-l)z:3 * where n = u(I ), by [5], Theorem 7.1. But Pk*(z) = A (l _22)—(r—2) P:(z) [24], Cor. 3.4.2, which equals (l-22)_(r-2)(l+-z)d. Thus P k(z) = (1+2)d A/I (1-22)r-2(1-n22-(n-1)23) is rational. CHAPTER I I I SOME EXAMPLES A local ring (A,m,k) is said to be a Golod ring if all the Massey operations on H(K) vanish. See [24], Chapter 4. More generally, let f :(A,m,k) 4 (B,n,k) be a homomorphism of local rings over k. That is, a commu- tative triangle ‘ f . A 4 B k is given with f local. Following Avramov, [5], f is * a small homomorphism if the induced map f :TorA(k,k) 4 TorB(k,k) is injective. A homomorphism. f :(A,m,k) 4 (B,n,k) over k is then called a Golod homomorphism if equivalently (l) f is small and TorA(B,k) has trivial Massey products (2) n -I TorA(B,k) = O and B PA/PB — 1-z(PA(z)-l), where I TorA(B,k) denotes the kernel of the canonical augmentation TorA(B,k) 4 k. Since the definition of the Massey products is quite lengthy and condition (2) is the 40 41 one which will actually be used below, the interested reader is again referred to [24] or to [4]. NOtice that Whenever f :A 4 B is surjective, n -I TorA(B,k) automatically is 26330. It is easy to see that a composition of small homomorphisms is small, [5], Lemma 3.8. The next example shows that a composition of Golod homomorphisms need not be a Golod homomorphism, a fact which does not seem to have appeared in the literature. Example 3.1. Let (A,m,k) be a regular local ring of dimension d. Let I = (10.x), I a perfect ideal of grade two, x a non—zero divisor on .A/IO, with u(IO) = n. Then pdALA/IO) = 2 by definition. Let 1 on-vAn' +An4A4A/IO-*O x G :O‘* A v A 4 A/XA * 0 be minimal resolutions over A of A/IO and x/xA, res- pectively. The fact that the ranks in F are equal to n and n-1 is seen by tensoring with the quotient field of A, a flat extension. Consider the product complex F®G:o»An“l+A2n‘l»An+l»A+A/I—»o. Since the differential on F ® G is defined by degif d(f®g) = df e g+ (-1) f o dg, d(F®G) gm(F®G), as F and G were minimal to start with. Moreover, Hi(F®G) = Tor?(A/IO,A/xA) = O for i > 1, since 42 pdA(A/xA) = l [33], 18.C, Lemma 6. But H1(F®G) = Tor}:(A/IO,A/XA) can be computed as H1(G®A/IO), which is zero since x was chosen to be a non-zero divisor on A/IO. Therefore, F ® G is a minimal resolution of A/I. So the Betti numbers of .A/I as A-module are bO = l, bl=n+l,b2=2n-l, b3=n-1,b4=b5 =...= O, and Pi/IM) = 1+ (n+l)z+ (2n-l)zz+ (n-1)23. k (1+2)d by [5], Theorem NOW, PA/Io(z) = 3 , l-nzz-(n-l)z 7.1. Also, P (z) (1) Pk = A/Io _ (1+2)C1 A/I 1-22 (l-nzz-(n-l)z3)(l-22) if x E m2 modulo IO, and k P (z) (2) Pk = 'A/IO = [14-Z)d A/I 1+2 (l-nzz—(n-l)z3)(l+z) for x E m\m2 modulo I because x is a 0’ non-zero divisor on A/IO, [24], Cor. 3.4.2. Suppose A +.A/I were a Golod homomorphism. Then by definition, k P (z) k A (3) P (2) AA l-Z(P§/I(Z)-l) = (1+2)d l - (n+ 1)z2 - (2n- 1)z3 - (n-1)z4 k (1+2)C1 I In case (1), P (z) # A/I 1_(n+]_)z2-(2n-1)z3-(r1--l)z4 since the denominators have different degrees. In case (2), a simple multiplication shows that 43 (l-I-z)d k P (z) 7! A/I l-(n+1)22--(21’1"]-)Z3‘”1"“z 4 , as required by (3). Thus the composition A 4 A/IO 4 A/I is not a Golod homomorphism, whereas both A 4 A/IO and A/IO 4 A/I are [5], Theorem 7.1 and [30], Theorem 3.7. Remark. This example was first considered by Buchsbaum and Eisenbud in [7], albeit for a different purpose. In [17], Fr5berg exhibited examples of two local Artin rings, one Gorenstein the other not, with the same Poincare series. Thus PX, or equivalently the deviations ei(A), cannot be used to characterize the class of Gorenstein local rings, something which is of course possible for regular rings and complete intersections. In the next two examples, the deviations are computed for two local rings, one Cohen-Macaulay the other not, which shows that such a homological characterization does not seem possible in this case either. Example 3.2. Let R = k[[X,Y,Z]], A = R/m, where T = (XY,YZ). Now, dim(A) = 2 and it is not hard to see that depth(A) = depthR(A) g_l. Since depthR(A) g dimR(A) = dim(A) = 2 < depthR(R), from the exact sequence of R-modules O 4 m 4 R‘4 A44 0 it follows that depthR(A) = depthR(fl)-l by [18], p.237. So depthR(A) g_1 if and only if depthR(M) g_2. But depthR(fl) # 3, the maximum possible value, since M would then be free as 44 R-module [33], p.113. Therefore, A is not Cohen-Macaulay. This can also be seen geometrically, since the variety k[X,Y,Z]/KXY,YZ) has an embedded component of dimension one through the origin. See [33], Theorem 30. However, =[13-z)2 l--z--z2 A is a Golod ring with P (z) A [24], Theorem 4.3.4 (A is clearly not a complete intersection). Once PA is known to be rational, the deviations can be computed using the d—invariants of Castillon and Micali [10]. Let g(z) = PA(-z). Then the d-invariants are the coefficients of the formal power series €365?” and one has the formula 6 m = l_—I]r‘l—)— Z “(IE)Q d|m d d’ m-l where u here denotes the Mobius function. For the example , 2 under consideration, H—éf%-= z '34 which is g (l+z-z)(l—z) -3 + l + 1 Y1 =li;%4£5 Y2 = 1 -345 upon l-z yl-z yz-z’ 2 ’ expanding by partial fractions. From this, a = -3, a = l, _ _ n 2 2 d3 - -6,.... an — —2+-(-l) (Yli-YZ). Hence, 60 61(A) = 11(91)= 2. 62(A) = l. 63(A) = l. 64(A) = 2. es( with the sequence monotone increasing from there on. Example 3.3. Let S = k[[X,Y]]. B = S/KX.Y)2. Since dim(B) = O, B is Cohen-Macaulay. B is however, not Gorenstein because O = (X2,Y) fl (X,Y2). On the other hand, B is a Golod ring [24], Theorem 4.3.5. Thus ) - (l+—z)2 where c = dim.H (K) Z ‘ 2 3 ' 1 l ' l-clz -c22 PB( 45 c2 = dim H2(K), K being the Koszul complex over B [24], Cor. 4.2.4. NoW dim Hl(K) = u((X,Y)2) = 3 and dim H2(K) = 2, since H2(K) a ann(m), where m is the maximal ideal of2 B [24], Lemma 1.4.2. Therefore, P (z) = (13-2) = l . P can be computed at least B 2 3 1-22 B l-3z -22 two other ways. B is a complete intersection modulo its socle [23], and B satisfies the hypotheses of Froberg's _ _ 1 .3;151._ ‘2 = _ n+1 n+1 _ 9(2) " 1+ 22' Hence an ( 1) 2 and 60(B) - 2, 81(3) = 3, €2(B) = 2, 53(B) = 3, 84(5) = 6, with the sequence monotone increasing thereafter. BIBLIOGRAPHY [l] [2] [3] [4] [5] [6] [7] [8] [9] [10] BIBLIOGRAPHY S.S. Abhyankar, "Local rings of high embedding dimension," Amer. J. Math. 89 (1967), 1073-1077. E.F. Assmus, "On the homology of local rings," Ill. J. Math. 3 (1959). 187-199. MmF. Atiyah and I.G. Macdonald, "Introduction to Commutative Algebra," Addison-Wesley, Reading, Mass., 1969. L.L. 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