ON SIMPLICIAL COMMUTATIVE ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY JAMES M. TURNER Abstract.In [22, 4] a conjecture was posed to the effect that if R ! A i* *s a homo- morphism of Noetherian rings then the Andr'e-Quillen homology on the cat* *egory of A-modules satisfies: Ds(A|R; -) = 0 for s 0 implies Ds(A|R; -) = 0 for* * s 3. In [28], an extended version of this conjecture was considered for which A * *is a simplicial commutative R-algebra with Noetherian homotopy such that char(ß0A) 6= 0.* * In ad- dition, a homotopy characterization of such algebras was described. The * *main goal of this paper is to develop a strategy for establishing this extended conje* *cture and provide a complete proof when R is Cohen-Macaulay of characteristic 2. Overview In [22], D. Quillen presented his viewpoint on the homology of algebras which* * ex- tended, in the commutative case, the work of Lichtenbaum and Schlessinger and g* *ave M. Andr'e's notion of homology. Furthermore, he observed that strong vanishing * *of this Andr'e-Quillen homology for finite type algebras held only when such algebras p* *ossessed the complete intersection property and conjectured that a weaker type of vanish* *ing also characterized such algebras. In [4], L. Avramov clarified and extended Quillen'* *s conjec- tures in the following manner. Let f : R ! A be a homomorphism of Noetherian ri* *ngs [Note: unless otherwise noted, all rings and algebras from this point on are co* *mmutative with unit]. Then f is a locally complete intersection provided for each q 2 Spe* *cS the semi-completion R"\R ! A^" suitably factors through a surjection with kernel be* *ing generated by a regular sequence (see below for more details). Quillen's Conjecture: (see [4, 22]) Let R ! A be a homomorphism of Noetherian rings such that the Andr'e-Quillen homology satisfies Ds(A|R; -) = 0 (as functo* *rs of A-modules) for s 0. Then (1) Ds(A|R; -) = 0 for s 3; (2) if fdR A < 1 then R ! A is a locally complete intersection (and, hen* *ce, Ds(A|R; -) = 0 for s 2). ___________ Date: July 9, 2003. 1991 Mathematics Subject Classification. Primary: 13D03; Secondary: 13D07, 13* *H10, 18G30, 55U35. Key words and phrases. simplicial commutative algebras, Andr'e-Quillen homolo* *gy, homotopy operations. Partially supported by National Science Foundation (USA) grant DMS-0206647 an* *d a Calvin Re- search Fellowship. He thanks the Lord for making his work possible. 1 2 JAMES M. TURNER Part 2 of this conjecture was proved by Avramov in [4]. Part 1 was proved by Av* *ramov and S. Iyengar for algebra retracts in [6]. Following ideas of Haynes Miller, an alternate approach to proving this conje* *cture was taken in [26, 27] when R is a field by viewing it as a special case of an algeb* *raic version of a theorem of J.P. Serre [25]. Following this line of thinking, in [27, 28] the * *more general consideration of Noetherian algebras was extended to simplicial commutative alg* *ebras with Noetherian homotopy, that is, simplicial commutative algebras A such that * *ß0A is Noetherian and ß*A is a finite graded ß0A-module. In using Andr'e-Quillen homol* *ogy to analyse such, we can use the type of tools first clarified by Andr'e and Qui* *llen: flat base change, transitivity sequence, localization etc. Cf. [1, 22, 28]. A partic* *ularly useful method for analysing simplicial commutative algebras in our present context thr* *ough homology is the following generalization of the main result in [5], proved in [* *28]. For each " 2 Spec(ß0A) the simplicial commutative algebra A0= A Li0A"(ß0A)"there i* *s a complete local ring R0and a homotopy commutative diagram '' R -! A OE # # _ ''0 0 R0 -! A with the following properties: (1) OE is a flat map and its closed fibre R0="R0is weakly regular; (2) _ is a D*(-|R; k("))-isomomorphism; (3) j0induces a surjection j0*: R0! (ß0A0, k(")) of local rings; (4) fdR(ß*A) finite implies that fdR0(ß*A0) is finite We call such a diagram a homotopy factorization of A. We can use such factoriz* *a- tions to extend the notion of locally complete intersection to simplicial commu* *tative R-algebras with Noetherian homotopy. Specifically, we call such A a a locally * *homo- topy n-intersection, n a natural number, provided for each " 2 Spec(ß0A) there * *is a factorization such that the connected component at " satisfies A(") := A0 LR0k(") ' Sk(")(W ) with W a connected simplicial k(")-module satisfying ßsW = 0 for s > n. Here a* *nd throughout Sk(")(-) denotes the free commutative k(")-algebra functor. We can now state, inspired by Serre's theorem [25], our simplicial version of* * Quillen's conjecture: Vanishing Conjecture: Let R be a Noetherian ring and let A be a simplicial comm* *uta- tive R-algebra with finite Noetherian homotopy and char(ß0A) 6= 0 such that the* * Andr'e- Quillen homology satisfies Ds(A|R; -) = 0 (as functors of ß0A-modules) for s * *0. Then (1) A is a locally homotopy 2-intersection; (2) if fdRß*A < 1 then A is a locally homotopy 1-intersection. ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 3 Part 2 of the Vanishing Conjecture was proved in [27, 28]. The goal of the fi* *rst part of this paper is to outline a strategy for giving a proof for the whole Vanishing * *Conjecture. The strategy involves formulating a more local version of the Algebraic Serre T* *heorem proved in [27] used to prove part 2 of the Vanishing Conjecture. Specifically,* * we will analyse the behavior of homotopy operations on each A("), particular the divide* *d pth- powers and the Andr'e operation (so named because of the role it played in [3] * *which motivated the direction of this paper). In the first section, we will formulate* * a Nilpotence and Non-nilpotence Conjecture regarding the action of these operations which, w* *hen coupled together, imply the Vanishing Conjecture. The second section will then * *focus on proving the Nilpotence Conjecture at the prime 2. Finally, in the third sect* *ion we will establish what will hopefully be our first step toward proving the Vanishing Co* *njecture when char(ß0A) = 2. Specifically, we will establish our: Main Theorem: Let A be a simplicial commutative R-algebra with finite Noetherian homotopy such that R is Cohen-Macaulay of characteristic 2. Then Ds(A|R; -) = 0* * (as a functor of ß0A-modules) for s 0 if and only if A is a locally homotopy 2-in* *tersection. As an immediate consequence, we obtain: Corollary. Let R ! A be a homomorphism of Noetherian rings of characteristic 2 * *such that R is Cohen-Macaulay. Then Ds(A|R; -) = 0 for s 0 implies Ds(A|R; -) = 0 * *for s 3. Acknowledgements. The author would like to thank Lucho Avramov for educating him on Cohen-Macaulay rings and for comments and criticisms on an earlier draft* * of this paper. He would also like to thank Paul Goerss for several discussions on * *homotopy operations as well as for many other helpful comments. 1.Nilpotence Conjectures In this section we reformulate the Vanishing Conjecture in terms of a two par* *t Nilpo- tence Conjecture which shifts the burden for global vanishing of Andr'e-Quillen* * homology to local vanishing of operations acting on the homotopy of components. We will* * first need a weaker notion of homotopy factorization in order to tighten our grip on * *how on how information from the homotopy of our simplicial algebra is transferred to t* *he homo- topy of its components. We will, throughout this section, be assuming basic pro* *perties of Andr'e-Quillen homology, refering the reader to [28] for details. 1.1. Weak homotopy factorizations. In the next subsection, we will recall that * *the conclusions of the Vanishing Conjectures are equivalent to certain strong globa* *l vanishing properties of Andr'e-Quillen homology. Our goal at present is to modify the not* *ion of homotopy factorizations which suitably preserves the Andr'e-Quillen homology bu* *t puts a tighter control on the local ring R0. Let A be a simplicial commutative R-algebra and denote ß0A by . We may assume that A is a simplicial commutative -algebra. Cf. [28, Theorem A]. Fix " 2 Sp* *ec 4 JAMES M. TURNER and let d(-)denote the completion functor on R-modules at ". Define the homoto* *py connected simplicial supplemented b-algebra A0by A0= A L b. Proposition 1.1. Suppose A is a simplicial commutative R-algebra with R a Noeth* *erian ring. Then there exists a (complete) local ring (R00, m), a simplicial commuta* *tive R00- algebra A00, and a homotopy commutative diagram '' R -! A (1.1) OE # # _ ''00 00 R00 -! A with the following properties: (1) OE is a complete intersection at m; (2) depth(m) = 0; (3) D 2(A|R; k(")) ~=D 2(A00|R00; k(")); (4) j00induces a surjection of local rings j00*: R00! ß0A00; (5) If A has finite Noetherian homotopy then A00has finite Noetherian homoto* *py. Proof: Choose a homotopy factorization of A over " '' R -! A OE # # _ ''0 0 R0 -! A which exists by [28, (2.8)]. Next, let q be the maximal ideal of R0. Let x1, . .,.xr be a maximal R0-subse* *quence of a minimal generating set for q. We define R00= R0=(x1, . .,.xr). Then m = q=(x1, . .,.xr)q has depth 0 since it contains only zero divisors. Fur* *thermore, the composite R" ! R0! R00is a complete intersection at m by definition. Cf. [4* *]. Now, let A00= A0 LR0R00. Then D 2(A|R; k(")) ~=D 2(A0|R; k(")) ~=D 2(A0|R0; k(")) ~=D 2(A0 LR0R00|R00; k("* *)) which follows from the properties of homotopy factorizations, the transitivity * *sequence, and flat base change [28, (2.4)]. Applying ß0 to the map R00! A0 LR0R00gives th* *e map R00~=R0 R0R00! ß0(A0) R0R00which is a surjection. Thus R00! ß0A00is a surjecti* *on. Finally, if A has finite Noetherian homotopy then so does A0(since ß*A0~=ßd*A* *). By [21, xII.6], there is a Kunneth spectral sequence 0 0 00 00 E2s,t= TorRs(ßtA , R ) =) ßs+tA . Since R0 ! R00is a complete intersection, fdR0R00< 1. Thus ß*A00will be a fini* *te module over ß0A00~=b R0R00. 2 ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 5 We will call a diagram (1.1) satisfying the conditions (1) - (5) above a weak* * homotopy factorization for A. 1.2. Brief review of the homotopy of simplicial commutative algebras over a field. Let A be a simplicial commutative `-algebra where ` is a field. In this * *section we review some basic facts about the homotopy groups of such objects, computed as * *the homotopy groups of simplicial `-modules. Let A` be the category of supplemented `-algebras, i.e. commutative `-algebra* *s aug- mented over `. Let sA` be the category of simplicial objects over A`. Then for * *A 2 sA` and n 0 we have a natural isomorphism ßnA ~=[S`(n), A]Ho(sA`) where S`(n) = S`(K(n)), K(n) the simplicial `-module satisfying ß*K(n) ~=` conc* *en- trated in degree n. We will use this relation to determine the natural primary * *algebra structure on ß*A. Given integers r1, . .,.rm , t1, . .,.tn 6= 0 an multioperation of degree (r1, . .,.rm ; t1, . .,.tn) is a natural map ` : ßr1x . .x.ßrm ! ßt1x . .x.ßtn of functors on sA`. Let Natr1,...,rm;t1,...,tnbe the set of multioperations of * *degree (r1, . .,.rm ; t1, . .,.tn). It is straightforward to show that Natr1,...,rm;t1,...,tn~=Natr1,...,rm;t1x . .x.Natr1,...,rm;tn. Now, we define (1.2) f : Natr1,...,rm;t! ßt(S`(r1) `. . .`S`(rm )) as follows. Let N = Natr1,...,rm;tand let X = S`(r1) `. . .`S`(rm ). For each 1* * j m, let 'j 2 ßrjX be the homotopy class of the inclusion S`(rj) ! X. Given ` 2 N th* *ere is an induced map `X : ßr1X x . .x.ßrmX ! ßtX. Thus we can define f : N ! ßtX by (1.3) f(`) = `X ('1, . .,.'m ). Proposition 1.2. Natr1,...,rm;t~=ßt(S`(r1) `. . .`S`(rm )). Proof: Since we have ßr1x . .x.ßrm ~=[S`(r1) `. . .`S`(rm ), -]Ho(sA`) the result follows from Yoneda's lemma [18]. * * 2 Note: (1) There is an obvious map Natr1,...,rm;tx Natt;q! Natr1,...,rm;q induced by composition. (2) Nat is naturally an `-module and f is naturally a linear map. 6 JAMES M. TURNER We now can address the issue of understanding possible relations among multio* *pera- tions. Corollary 1.3. For ` 2 Natr1,...,rm;tthen any expression for ` in Natr1,...,rm;* *qis deter- mined by f(`) 2 ßt(S`(r1) `. . .`S`(rm )). Furthermore, if _ 2 Natt;qthen f(_ * *O `) = f(_)Of(`), as composites of their homotopy representitives, in ßt(S`(r1) `. . .* *`S`(rm )). Proof: This again follows from Yoneda's lemma [18]. * * 2 Now, we are in a position to determine the full natural primary structure for* * homotopy in sA`. First, recall that for any field F we have (1.4) SF(V W ) ~=SF(V ) SF(W ). Next, let k = Q if char(`) = 0 and let k = Fp if char(`) = p 6= 0. We seek a na* *tural map of `-algebras OEV : S`(V k `) ! Sk(V ) k ` where V is a k-module. This can be defined as the adjunction of the inclusion V* * k` ! I(Sk(V ) k `) (here I : A` ! V` is the augmentation ideal functor). Proposition 1.4. The natural map OE : S`((-) k `) ! Sk(-) k ` is an isomorphi* *sm of functors from k-modules to A`. Proof: By the identity (1.4) and naturality, it is enough to provide a proof f* *or one dimensional V , i.e. for V ~=k. Then OEV : `[x] ! k[x] k` is determined alg* *ebraically by the value OEV (x) = x k 1. This is clearly an isomorphism. * * 2 Note: This and other similar results can also be shown to follow from the fait* *hful flatness of the functor (-) k `. Corollary 1.5. For V 2 sVk there is a natural isomorphism ß*(S`(V k `)) ~=ß*(Sk(V )) k `. As a consequence all natural primary homotopy operations for simplicial supplem* *ented `-algebras and their relations are determined by ß*Sk(n) for all n 2 N. Proof: The first statement follows from Proposition 1.4 and the faithful flatn* *ess of (-) k`. The second statement follows additionally from Corollary 1.3 and the Ku* *nneth theorem. Recall that S`(n) ~=S`(K`(n)) and we can take K`(n) = ` ~=k k`, where Sn is a choice of simplicial set model for the n-sphere. * * 2 Note: The computation of ß*SQ(n) can be traced back to at least as early as [9]* *. The computation of ß*SFp(n) can be found in [8, 9], for general p, and in [11] for * *p = 2. We will review the results of [11] in the next section. For non-zero characteristics, we will be interested in two particular operati* *ons. Specif- ically, for A 2 sA`, ß*A is naturally a divided power algebra. Therefore, there* * is a divided pth-power operation flp : ßnA ! ßpnA. ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 7 Cartan, Bousfield, and Dwyer also construct an operation # : ßnA ! ß(p-1)n+1A which we call the Andr'e operation because of the role it played in [3] where M* *. Andr'e's showed that Gulliksen's result about the equality of deviations with simplicial* * dimensions for rational local rings [15] cannot be extended to the primary case. In the no* *tion of [8, 8.8] and [11], ( ffin-1 p = 2, (1.5) # = (n-1)=2 p > 2. A useful basic relation between the two operations is (1.6) #flp = 0 1.3. Nilpotency conjectures and consequences. We now are in a position to ad- dress the Vanishing Conjecture and reformulate it in terms of local conditions * *on homo- topy groups. To begin, we need the following Lemma 1.6. Let W 2 sV`, with char(`) > 0, and let n 2 N be so that ßjW 6= 0 imp* *lies n j 1. Then (1) flp = 0 on ß*S`(W ) provided n = 1; (2) # = 0 on ß*S`(W ) provided n = 2. Proof: By Corollary 1.5, it is enough to provide a proof for ` = Fp. For n = 1,* * ß*S`(W ) is a free exterior algebra generated by ß1W , which has trivial flp-action. Fo* *r n = 2, ß*S`(W ) is a free divided power algebra generated by ß*W . Cf. [9]. Thus ß*S`(* *W ) has trivial #-action by relation (1.6). * * 2 Given A 2 sA` with char(`) > 0, we call A (1) -nilpotent provided flsp(x) = 0 for s 0 for all x 2 ß*A and (2) Andr'e nilpotent provided #s(x) = 0 for s 0 for all x 2 ß*A. Next, given a Noetherian ring R and a simplicial commutative R-algebra A with N* *oe- therian homotopy. For " 2 Spec(ß0A) with char(k(")) > 0, we call A (1) -nilpotent at " provided there is a homotopy factorization at " such th* *at A(") is -nilpotent over k("), and (2) Andr'e nilpotent at " provided there is a weak homotopy factorization at* * " such that A00 R00k(") is Andr'e nilpotent over k("). Proposition 1.7. Let A be a simplicial commutative R-algebra with Noetherian ho* *mo- topy and " 2 Spec(ß0A) such that char(k(")) > 0. Then (1) A is -nilpotent at " provided A is a homotopy 1-intersection at "; (2) A is Andr'e-nilpotent at " provided A is a homotopy 2-intersection at ". Proof: Both follow from the definitions and Lemma 1.6. * * 2 We now can state our two nilpotence-type conjectures. 8 JAMES M. TURNER Nilpotence Conjecture: Let A be a simplicial commutative R-algebra with finite * *Noe- therian homotopy. Let " 2 Spec(ß0A) be such that char(k(")) > 0 and Ds(A|R; k("* *)) = 0 for s 0. Then: (1) A is -nilpotent at " if and only if A is a homotopy 1-intersection at "; (2) A is Andr'e nilpotent at " if and only if A is a homotopy 2-intersection* * at ". Non-Nilpotence Conjecture: Let A be a simplicial commutative R-algebra with fi- nite Noetherian homotopy. Let " 2 Spec(ß0A) be such that char(k(")) > 0. Then Ds(A|R; k(")) 6= 0 for infinitely many s 2 N provided A fails to be Andr'e nilp* *otent at ". Remark: A motivation for the Nilpotence Conjecture came from dual topological r* *esults centered around conjectures of Serre and Sullivan as addressed in [20, 16, 14].* * See also [26] for further speculations on formulating other dual results. Assuming both of these conjectures, we can now provide a: Proof of the Vanishing Conjecture: First, we have Ds(A|R; k(")) = 0 for s 0. * *Since char(ß0A) > 0 then char(k(")) > 0. Thus, by the Non-Nilpotence Conjecture, A is Andr'e nilpotent at ". By the Nilpotence Conjecture, A is a homotopy 2-interse* *ction. If, additionally, fdR(ß*A) < 1 it follows that ß*(A0 R0k(")) is finite and, hen* *ce, - nilpotent. It follows from the Nilpotence Conjecture that A is a homotopy 1-int* *ersection at ". 2 2.Proof of the Nilpotence Conjecture at the prime 2 The goal of this section will be to provide a proof of the Nilpotence Conject* *ure when the base field has characteristic 2. This will involve a careful study of a ce* *rtain map, the character map, defined on the homotopy of of simplicial supplemented algebr* *as with finite Andr'e-Quillen homology, whose non-triviality implies the Nilpotence Con* *jecture. In fact, in the process of analysing this character map, we will be able to est* *ablish an upper bound on the top non-trivial degree of the Andr'e-Quillen homology in ter* *ms of the non-nilpotence of certain operations acting on homotopy. Along the way we will * *review some results from [11] and [12] and generalize them to arbitrary fields of char* *acteristic 2. 2.1. Connected envelopes and the character map. We close this section by provid- ing a strategy for proving the Nilpotence Conjecture. This will involve first r* *eviewing the concept of connected envelopes from [27]. We then construct the notion of a cha* *racter map for connected simplicial supplemented algebras with finite Andr'e-Quillen h* *omol- ogy and state a conjecture regarding this map whose validity implies the Nilpot* *ence Conjecture. Given A in sA`, which is connected, we define its connected envelopes to be a sequence of cofibrations j1 j2 jn-1 jn A = A(1) ! A(2) ! . . .! A(n) ! . . . ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 9 with the following properties: (1) For each n 1, A(n) is a (n - 1)-connected. (2) For s n, HQsA(n) ~=HQsA. (3) There is a cofibration sequence fn jn S`(HQnA, n) ! A(n) ! A(n + 1). Here we write, for B 2 sA`, HQs(A) := Ds(A|`; `) and, for V 2 V`, S`(V, m) := S`(K(V, m)). Existence of connected envelopes is proved in [27, x2]. Note: Paul Goerss has pointed out that connected envelopes can also be construc* *ted through a "reverse" decomposition via collapsing skeleta on the canonical CW ap* *proxi- mation. Now, for A 2 sA` connected, define the Andr'e-Quillen dimension of A to be AQ-dim (A) = max {m 2 N | HQm(A) 6= 0}. Assume that n = AQ-dim A < 1. Then A(n) ' S`(HQn(A), n). Cf. [27, (2.1.3)]. Summarizing, we have Proposition 2.1. For A 2 sA` connected and AQ-dim A < 1 there is a natural map OEA : A ! S`(HQn(A), n), where n = AQ-dim A, with the property that HQn(OEA) is an isomorphism. Now, assuming char(`) > 0, we noted that ß*B is naturally a divided power alg* *ebra. Given a divided power algebra in characteristic p, let J to be the divide* *d power ideal generated by all decomposables w1w2. .w.rand flp(z) with w1, w2, . .,.wr,* * z 2 1. Define the -indecomposables to be Q = =J. Given A 2 sA` connected and n = AQ-dim A finite, we define the character map of* * A to be A = Q ß*(OEA) : Q ß*A ! Q ß*(S`(HQn(A), n)). Now, for B 2 sA`, the action of the Andr'e operation # on ß*B induces an acti* *on on Q ß*B by the relation (1.6) and the fact that # kills decomposables of eleme* *nts of positive degree. Cf. [8, (8.9)]. Theorem 2.2. Let A 2 sA` be connected with char(`) = 2 and HQ*(A) a non-trivial finite graded `-module. Then A is non-trivial. 10 JAMES M. TURNER Proof of Nilpotence Conjecture at the prime 2: Let n = AQ-dim B where B = A00 R* *00` with ` = k("). By Corollary 1.5 and [12, (3.5)], # acts non-nilpotently on ever* *y non- trivial element of Q ß*(S`(HQn(B), n)) if n 3. Therefore if ß*B is Andr'e nil* *potent then Theorem 2.2 implies that n 2. Thus B is a homotopy 2-intersection by [27, (2.* *2)]. Since HQ*(B) ~=HQ*(A(")), if A is additionally -nilpotent at " then A is a h* *omotopy 1-intersection at ", as ß*A(") is free as a divided power algebra. * * 2 The goal of the rest of this section will be to provide a proof of Theorem 2.* *2. Remark: If A is a simplicial commutative R-algebra with finite Noetherian homot* *opy such that char(ß0A) = 2, then, for each " 2 Spec(ß0A), char(k(")) = 2. Thus The* *orem 2.2 coupled with the Non-Nilpotence Conjecture implies (1) of the Vanishing Con* *jecture when char(ß0A) = 2. Furthermore, as an inspection of the proof of the Vanishing* * Con- jecture above shows, (2) of the Vanishing Conjecture follows directly from the * *Nilpotence Conjecture. Thus we have an alternative proof of [28, Theorem B] when char(ß0A)* * = 2. 2.2. Review of homotopy operations in characteristic 2. Let A be a simplicial commutative algebra of characteristic 2 (and, therefore, a simplicial F2-algebr* *a). Asso- ciated to A is a chain complex, (C(A), @), where, for each n 2 N, we have C(A)n = An, @ = ni=0di: C(A)n ! C(A)n-1. It is standard that we have the identity [17] ßnA ~=Hn(C(A)). In [11], W. Dwyer showed the existence of natural chain maps k : (C(V ) C(W ))i+k! C(V W )i 0 k i, where V and W are simplicial F2-modules, having the following properties: (1) 0 + T 0T = + OE0; (2) k + T kT = @ k-1+ k-1@. Here T : C(V ) C(W ) ! C(W ) C(V ) is the twist map, : C(V ) C(W ) ! C(V W ) is the shuffle map [17, p. 243], and OEk : C(V ) C(W ) ! C(V W ) is the deg* *ree (-k) map defined by ( 0 degv 6= k ordeg w 6= k; OEk(v w) = v w otherwise. Note: Tensor product of chain complexes is graded tensor product and tensor pro* *duct of simplicial modules is levelwise tensor product. Now, for x 2 C(A)n and 1 i n, define i(x) 2 C(A)n+iby i(x) = ffn-i(x) w* *here fft(x) = ~ t(x x) + ~ t-1(x @x), and ff0(x) = ~ 0(x x), ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 11 where ~ is the map C(A A) ! C(A) induced by the product on A. As shown in [12, x3], these natural maps have the following properties: (1) @ i(x) = i(@x) for 2 i n; (2) @ n(x) = ~ (x @x); (3) @ 1(x) = 1(@x) + x2; ( @~ n-i-1(x y) 2 i < n; (4) i(x + y) = i(x) + i(x) + ~ (x y) i = n. From these chain properties for the i, there are induced homotopy operations ffii: ßnA ! ßn+iA 2 i n, or, upon letting fft= ffin-t, we have fft: ßnA ! ß2n-tA 0 t n - 2. Note, in particular, that (2.7) # = ff1. The following is proved in [11, 13]: Theorem 2.3. The homotopy operations ffii have the following properties: (1) ffii is a homomorphism for 2 i n - 1 and ffin = fl2 - the divided sq* *uare; (2) ffii acts on products as follows: 8 >: 0 otherwise; (3) if i < 2j, then X ` j - i + k -'1 ffiiffij = ffii+j-kffik. i+1_2 k i+j_3j - s Corollary 2.4. The homotopy operations fft have the following properties: (1) fft is a homomorphism for 1 i n - 2 and ff0 = fl2 - the divided squa* *re; (2) fft acts on products as follows: 8 >: 0 otherwise; (3) if s > t, then X ` s - q - 1' ffsfft= ffs+2t-2qffq. s+2t_3 q s+t-1_2q - t 12 JAMES M. TURNER Proof of Corollary 2.4: The first two items follow immediately from Theorem 2.3* * using the identity fft(x) = ffin-t(x) where deg x = n. The last relation follows fro* *m (3) of Theorem 2.3 upon letting j = n - t, i = 2n - s - t, and k = n - q. * * 2 Our goal at present is to describe homotopy operations for simplicial algebra* *s over general fields ` of characteristic 2. Specifically, we will prove: Theorem 2.5. Let A be a simplicial supplemented `-algebra with char(`) = 2. Th* *en, for 2 i n, the natural operation ffii : ßnA ! ßn+iA satifies properties (1)* * - (3) of Theorem 2.3. In particular, for a, b 2 ` and x, y 2 ßnA we have ( (ab)(xy) i = n; ffii(ax + by) = a2ffii(x) + b2ffii(y) + 0 otherwise. and for u, v 2 ß*A 8 ><(ab)2(ffii(u)v2)degv = 0; ffii((au)(bv)) = (ab)2(u2ffii(v))degu = 0; >: 0 otherwise. Furthermore, homotopy operations ßnA ! ßn+kA, as natural maps of functors of si* *m- plicial supplemented `-algebras, are determined algebraically over ` by the ope* *rations ffii1ffii2. .f.fiirwith (i1, . .,.ir) an admissible sequence of degree k and ex* *cess n. Recall that the degree of I = (i1, . .,.ir) is i1 + . .+.Ir and the excess of* * I is i1 - i2 - . .-.ir. We will write throughout ffiI = ffii1ffii2. .f.fiir. Finally, we * *call I admissible provided iq-1 2iq for all 2 q r. To prove Theorem 2.5, we need two lemmas. Lemma 2.6. For n 1, we have ß*SF2(n) ~= [ffiI('n)| excess(I) < n]. It follows that for any field ` of characteristic 2 ß*S`(n) ~= `[ffiI('n)| excess(I) < n]. Proof: For the first statement, see [8, x7] or [11, Remark 2.3]. The second st* *atement follows from the first and Proposition 1.4. * * 2 For the following, see [12, 12.4.2]. Lemma 2.7. Let A and B be simplicial commutative F2-algebras. Then the induced action of ffii on ß*(A) F2ß*B is determined by 8 >: 0 otherwise. ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 13 Proof of Theorem 2.5: Since ` has characteristic 2, the operations ffiiare defi* *ned on ßnA and satisfy (1) through (3) of Theorem 2.3. In particular, to compute ffii(ax +* * by) it is enough, by Corollary 1.5, to compute ffii(a'n `1 + 1 `b'n) 2 ß*(S`(n)) `ß*(S`(n)). Under the isomorphism (using Proposition 1.4 and Kunneth Theorem) ß*(S`(n)) `ß*(S`(n)) ~=(ß*(SF2(n) F2ß*(SF2(n))) F2`, ffii(a'n `1 + 1 `b'n) corresponds to ffii(('n F21) F2a + (1 F2'n) F2b). T* *hus the desired result follows from Lemma 2.7. Similarly, to compute ffii((au)(bv)) it * *is enough to compute ffii((a'm ) `(b'n)) 2 ß*(S`(m)) `ß*(S`(n)), or, equivalently, ffii(('* *m F2'n) F2 (ab)) 2 (ß*(SF2(m)) F2ß*(SF2(n))) F2`. This again can be computed using Lemma 2.7. Finally, the last statement follows from Corollary 1.5 and Lemma 2.6. * * 2 Note: Theorem 2.5 shows that the operations ffiiand the relations (1) - (3) of * *Theorem 2.3 completely determine the homotopy operations for simplicial supplemented al* *gebras over general fields of characteristic 2. Thus the Galois group of ` over F2 pro* *duces no new homotopy operation of positive degree nor alters the relations between them* *. This should not be surprising as the same considerations is known to hold rationally* *. See [24, x4]. 2.3. Quillen's spectral sequence. We now modify the results of [12, x6] to enab* *le to use Quillen's fundamental spectral sequence [22, 23] over general fields of cha* *racteristic 2. To begin, we need to be more explicit about the functors S`(-). Let V be an `* *-module. For n 2 N, define S`,0(V ) = ` and S`,n(V ) = `. Then S`(V ) ~= n2NS`,n(V ). Next, let W be a non-negatively graded `-module and define (2.8) S`(W ) = `[ffiI(w) | w 2 W, I admissible, excess(I) < degw] which, by Corollary 2.4, can be expressed as (2.9) S`(W ) ~= `[ffi11ffi22. .f.fin-2n-2(w) | w 2 W, n = degw, i1, . .,.in* *-2 2 Z+]. For u 2 S`(W ), we define the weight of u, wt(u), as follows: 8 >>0 ifu 2 `; >< 1 ifu 2 W ; wt(u) = >>wt(x) + wt(y) ifu = xy; >: 2 wt(x) ifu = ffii(x). We then define, for n 2 N, S`,n(W ) = `. 14 JAMES M. TURNER Proposition 2.8. For a simplicial F2-module V and n 2 N there are a natural iso* *mor- phisms S`,n(V F2`) ~=SF2,n(V ) F2` and S`,n((ß*V ) F2`) ~=SF2,n(ß*V ) F2`. As a consequence, if W is a simplicial `-module then ß*S`,n(W ) ~=S`,n(ß*W ). Proof: The first two statements can be proved just as for Proposition 1.4. For * *the last statement, note that [12, x3] shows that the isomorphism holds when ` = F2. Not* *e also that a standard argument (e.g. via Postnikov towers) shows that there is a simp* *licial set X and a homotopy equivalence W ' `. Thus ß*S`,n(W ) ~=SF2,n(ß*V ) F2` where V = F2. Since ß*W ~=(ß*V ) F2` it follows that S`,n(ß*W ) ~=SF2,n(ß*V ) F2`. 2 We now follow [12, x6]. Let A be a simplicial supplemented `-algebra and let * *IA be its augmentation ideal. We may assume, using the standard model category structure * *[21, xII.3], that A is almost free, i.e. At ~=S`(Vt) for all t 1. Furthermore, the* * composite Vt IAt ! QAt to the indecomposables module is an isomorphism. We now form a decreasing filtration of A: Fs = (IA)s. For A almost free, E0sA = Fs=Fs+1= (IA)s=(IA)s+1~= S`,s(QA). Applying homotopy gives a spectral sequence (2.10) E1s,tA = ßtE0tA ~=ßtS`,s(QA) =) ßtA with differentials (2.11) dr : Ers,tA ! Ers+r,t-1. This is called Quillen's spectral sequence. Theorem 2.9. For a simplicial supplemented `-algebra A there is a spectral sequ* *ence of algebras E1As,t= S`,s(HQ*(A))t=) ßtA with the following properties: (1) The spectral sequence converges if ß0A ~=`. In particular, Ers,tA = 0 fo* *r t < s for all r 1. (2) For 1 r 1 there are operations ffii: Ers,tA ! Er2s,t+iA 2 i t of indeterminacy 2r-1 with the following properties: ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 15 (a)If r = 1, then ffii coincides with the induced operation S`,s(HQ*(A)* *)t ! S`,2s(HQ*(A))t+i. (b) If x 2 ErA and 2 i < t then ffii(x) survives to E2rA and d2rffii(x) = ffii(drx) drffit(x) = xdrx modulo indeterminacy. (c)The operations on ErA are induced by the operations on Er-1A and the operations on E1 A are induced by the operations on ErA for r < 1. (d) The operations on E1 A are induced by the operations on ß*A. (e)Up to indeterminacy, the operations on ErA satisfy the properties of* * Theo- rem 2.5. Before we indicate a proof of this omnibus result, a word of explanation is n* *eeded. First, an element y 2 Ers,tA is said to be defined up to indeterminacy q provid* *ed y is a coset representitive for a particular element of Ers,tA=Bqs,tA where Bqs,tA Ers,tA q r is the `-module of elements of Ers,tA which survive to Eqs,tA but have zero res* *idue class. Also, if A is almost free, and hence cofibrant as a simplicial supplemented `* *-algebra, then ß*(QA) ~=HQ*(A). Cf. [27, x1]. Proof: First, if A is almost free, we have a pairing (~ )* ßt(S`,s(QA)) ßt0(S`,s0(QA)) ! ßt+t0(S`,s+s0(QA)) which gives a pairing E1s,tA E1s0,t0A ! E1s+s0,t+t0A and induces an algebra structure on the spectral sequence. For (1), we simply note that if A is connected then S`,s(HQ*(A)) = 0 for t > * *s. Convergence now follows from standard convergence theorems. Cf. [17]. For (2), we have a commutative diagram C((IA)s) F2C((IA)s) -~fft!C((IA)s F2(IA)s) -! C((IA)s `(IA)s) oe " # ~ C((IA)s) -fft! C((IA)2s) = C((IA)2s) where oe(u) = u u and ~fft(a b) = t(a b) + t-1(a @b). This induces a * *map i: (IA)s ! (IA)2s, by again setting i(u) = ffn-i(u) where n = degu. Let x 2 Ers,tA. Then, modulo (IA)s+1, x is represented by u 2 (IA)s with the * *property that @u 2 (IA)s+r. The class of u is not unique, but may be altered by adding e* *lements @b 2 (IA)s with b 2 (IA)s-r+1. 16 JAMES M. TURNER Define ffii(x) 2 Er2s,t+iA to be the residue class of i(u). Since @ i(u) = i(@u) 2 (IA)2s+2r 2 i < t, and @ t(u) = ~ (u @u) 2 (IA)2s+r. Thus ffii(x) is defined in Er2s,t+iA and survives to E2rA with d2rffii(x) = ffi* *i(drx) for 2 i < t. Also drffit(x) = xdrx. This gives us (b). Now we have a commuting diagram ( i)* 2s ßt((IA)s) -! ßt+i((IA) ) # # ffii ßtA -! ßt+iA and an induced diagram ( i)* 2s ßt((IA)s) -! ßt+i((IA) ) # # ßt((IA)s=(IA)s+1) -! ßt+i((IA)2s=(IA)2s+1) #~= ~=# ffii Q S`,s(HQ*(A))t -! S`,2s(H* (A))t+i It is now straightforward to check (a), (c), (d), and (e). * * 2 2.4. Non-triviality of the character map. We now proceed to prove Theorem 2.2. We will in fact prove a more general theorem. Specifically: Theorem 2.10. Let A be a simplicial supplemented `-algebra (char(`) = 2) such t* *hat HQ*(A) is finite graded as an `-module. Let n = AQ-dim A and assume n 2. Th* *en there exists x 2 ß*A and y 6= 0 2 HQn(A) such that under the map ß*OEA : ß*A ! ß*S`(HQn(A), n) we have (ß*OEA)(x) = fftn-2(y) for some t 1. Proof of Theorem 2.2: Assume n = AQ-dim A 3. Let y 2 HQn(A), and x 2 ß*A sati* *sfy the properties of Theorem 2.10. By Equation 2.9, fftn-2(y) 6= 0 in Q ß*(S`(HQn(* *A), n)) for all t 1. We conclude that A(x) 6= 0. If n 2, then A is a surjection and, hence, non-trivial. * * 2 Now, in order to prove Theorem 2.10 we will need to know something about the annihilation properties of homotopy operations. Specifically, we will focus on * *composite operations of the form `(s, t) = ffi2sffi2s-1. .f.fi2t+1 s > t (where we set `(t + 1, t) = ffi2t+1). ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 17 Lemma 2.11. Let i 2 and t 1 be such that 2t< i. Then `(s, t)ffii= 0 for s * * t. Proof: Write i = 2t-1+ n with n 1. Note first that an application of the rel* *ation Theorem 2.3 (3) shows that for any t 1, ffi2t+1ffi2t+1= 0 = ffi2t+1ffi2t+2. We thus assume, by induction, that for any t and 0 < j < n, there exists s t * *such that `(s, t)ffi2t+j= 0. By another application of the relation Theorem 2.3 (3), we have X ` n + r - 1' ffi2t+1ffi2t+n= ffi2t+1+n-rffi2t+r. 1 r n_3 n - r Notice that, for each such r, 2t+1< 2t+1+ n - r < 2t+1+ n. Thus, by induction, * *we can find s t + 1 so that ` ' X n + r - 1 `(s, t + 1) ffi2t+1+n-rffi2t+r = 0. 1 r n_3 n - r We conclude that `(s, t)ffi2t+n= `(s, t + 1)ffi2t+1ffi2t+n= 0. 2 Corollary 2.12. Let I = (i1, . .,.ik) be an admissible sequence and let t < k. * * Then `(s, t)ffiI = 0 for s t. Proof: Since I is admissible, then i1 2i2 . . .2k-1ik 2k > 2t. Thus, by Lemma 2.11, `(s, t)ffiI = (`(s, t)ffii1)ffii2. .f.fiik= 0 for s t. 2 Proposition 2.13. Let A be a connected simplicial supplemented `-algebra, char(* *`) = 2. Let y 6= 0 in E11,nA ~= HQn(A), n 2. Then there exists s 1 such that ffsn-* *2(y) 2 E12s,n+2s+1-2A survives to E1 A (though possibly trivially). Proof: Choose m 1 and suppose ffmn-2(y) survives to ErA, r 1. By Theorem 2.* *9 (2) (a), we may assume that r 2m . Let w = dr([ffmn-2(y)]) 2 Er2m+r,n+2m+1-3A by * *(2.11). By Theorem 2.9 (1), w = 0 provided n + 2m - 2 r. Thus if r n + 2m - 2 then the class of ffmn-2(y) survives to E1 A as all subsequent differentials will sa* *tisfy the same criterion. Suppose next that r < n + 2m - 2. Write n + 2m - q = r with n q >m2. Assume* *, by induction, that if for some m the class of ffmn-2(y) survives to En+22-jm,n+2m+* *1-2A for q > j then there exists s m such that the class of ffsn-2(y) survives to E1 A. Aga* *in, let 18 JAMES M. TURNER w = dr([ffmn-2(y)]) 2 Er2m+r,n+2m+1-3A. Choose u 2 E12m+r,n+2m+1-3A to represe* *nt the class w. By Theorem 2.9 and Proposition 1.4, we have (P P aI,lffiI(xl) + JbJzJr = 2k - 2m , k > m; u = P I,l JbJzJ otherwise where I = (i1, . .,.ik) and J = (j1, . .,.jr) are sequences with I admissible, * *aI,k, bJ 2 `, and zJ = zj1zj2. .z.jr+2mwith xk, zj1, . .,.zjr+2m2 HQ*(A). First assume that r 6= 2k - 2m . Then dr([ffmn-2(y)]) = [u] 2 Er2m+r,n+2m+1-* *3A with u 2 E12m+r,n+2m+1-3A decomposable. Note again that deg u > 2m . Thus, by Theorem 2.9 (2) (b), (c), and (e), d2r(ffi2m+1[ffmn-2(y)]) = ffi2m+1dr([ffmn-2(y)]) = f* *fi2m+1[u] = 0. Thus [ffm+1n-2y] survives to E2r+12m+1,n+2m+2-2A. Now, let 2r + 1 = n + 2m+1 - j and* * recall that r = n + 2m - q 2m . Then j = (n + 2m+1) - (2n + 2m+1 - 2q) - 1 = 2q - n - 1 = q - (n - q) - 1 < q. Thus, by induction, there exists s m such that [ffsn-2(y)] survives to E1 . Now assume that r = 2k - 2m with k > m. By definitions of ffn-2 and `(m, t), ffmn-2(y) = `(m, 0)(y). By Theorem 2.9 (2) (b) and (c), for e > m d2e-mr([`(e, 0)y]) = `(e, m)dr([`(m, 0)(y)]) = `(e, m)w. By Theorem 2.9 (2) (c) and (e) and Theorem 2.5, `(e, m)w is represented by X e-m a2I,l`(e, m)ffiI(zl) modulo indeterminacy. I,l Note that 2m < deg u so we can assume there are no decomposables in our choice * *of representative for `(e, m)w. As indicated above, we have for each I = (i1, . .,* *.ik) that k > m. Thus, by Corollary 2.12, since the sum is finite, there exists e m suc* *h that `(e, m)ffiI = 0 for allI. Thus d2e-mr([`(e, 0)y]) = 0 modulo indeterminacy. Therefore [`(e, 0)y] survives* * to e-mr+1 s E22e,n+2e+1-2A, so, by the previous case, there exists s e such that [ffn-2(y* *)] = [`(s, 0)y] survives to E1 . 2 Proof of Theorem 2.10: Choose y 2 HQn(A) ~= E11,nA and choose s 1 such that ffsn-2(y) 2 E1A survives to E1 A, which exists by Proposition 2.13. Under the i* *nduced map Er(OEA) : ErA ! ErS`(HQn(A), n) we have Er(OEA)([ffsn-2(y)]) = [ffsn-2(y)] for all 1 r 1. But, since E1S`(HQn(A), n) ~=E1 S`(HQn(A), n), ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 19 we can conclude that E1 (OEA)([ffsn-2(y)]) 6= 0. Thus we can find a nontrivial * *x 2 ß*A which is represented by ffsn-2(y) in E1 A such that (ß*OEA)(x) = ffsn-2(y). * * 2 Remarks: (1) From the proof of Proposition 2.13, an algorithm can be made to determin* *e an s such that ffsn-2(x) survives to E1 A for x 2 HQn(A). Choose m 1 such* * that m+1 ffmn-2(x) survives to E2 A, guaranteed by Theorem 2.9.2 (b) and Coroll* *ary 2.12 (see also [12, (6.9)]). Then, using the procedure in the proof, it can * *be shown that ffm+n-2n-2(x) survives to E1 A. (2) Following the philosophy of [26], the reader can conjecture a dual topol* *ogical version of Theorem 2.10 for nilpotent finite Postnikov towers, using con* *nected covers, which would further generalize results of Serre from [25] in the* * spirit of [16, 14]. 3. Proof of the Main Theorem We now seek to establish a special case of the Vanishing Conjecture as descri* *bed in the overview. The proof will utilize the validity of the Nilpotence Conjecture* * at the prime 2 while avoiding the need to evoke the Non-Nilpotence Conjecture. Let A be a simplicial commutative F2-algebra and let (C(A), @) be the associa* *ted chain complex. The following is proved in [2, 11]. Proposition 3.1. The shuffle map : C(A) F2C(A) ! C(A F2A) induces a divided power algebra structure on C(A). Specifically, for each k 2 Z+, there is a fun* *ction flk : C(A)n ! C(A)kn satisfying: (1) fl0(x) = 1 and fl1(x) = x (2) flh(x)flk(x) =P h+khflh+k(x) (3) flk(x + y) = r+s=kflr(x)fls(x) (4) flk(xy) = 0 for k 2 and x, y 2 C(A) 1 (5) flk(xy) = xkflk(y) for x 2 C(A)0 and y 2 C(A) 2 (6) flk(fl2(x)) = fl2k(x) (7) @flk(x) = (@x)flk-1(x) (8) u 2 C(A)n a cycle then, for [u] 2 ßnA, ffin([u]) = [fl2(u)]. Let A ! B be a map of simplicial commutative F2-algebras and æ : C(A) ! C(B) the induced map of chain complexes. Then for u 2 C(A)n and all n > i 0 æ(ffi(u)) = ffi(æ(u)) where ffi= n-i. Recall (2.7) that # = ff1. Lemma 3.2. Let A ! B be a map of simplicial commutative F2-algebras and suppose ßsA = 0 for s 0. Let u 2 C(A)n, n 3, such that æ(@u) = 0. Then æ(u) is a cy* *cle in C(B) and #r([æ(u)]) = 0 in ß*(B) for r 0 provided flr2(@u) = 0 in C(A) for* * r 0. Proof: First, in C(A), we have, by an induction using the formulas for i from * *x2.2, that @#r(u) = flr2(@u). 20 JAMES M. TURNER Since flr2(@u) = 0 for r 0 and Hs(C(A)) = 0 for s 0, it follows that #r(u) * *is a boundary in C(A) for r 0. We conclude that #r([æ(u)]) = [æ(#r(u)] = 0 in ß*(B* *). 2 Corollary 3.3. Let A ! B be a level-wise surjection of simplicial commutative F* *2- algebras such that fl2 acts locally nilpotently on (@C(A)) \ keræ and ßsA = 0 f* *or s 0. Then B is Andr'e nilpotent. Proof: Given x 2 ßnB with n 3, let w 2 C(B)n be a cycle representitive for x * *and choose u 2 C(A) such that æ(u) = w. Then æ(@u) = 0 and flr2(@u) = 0 for r 0 by assumption. Thus #r(x) = 0 for r 0 by Lemma 3.2. 2 Proof of the Main Theorem: Let A be a simplicial commutative R-algebra with fi- nite Noetherian homotopy, R a Cohen-Macaulay ring of characteristic 2, such that Ds(A|R; -) = 0 for s 0, as a functor of ß0A-modules. Note that A has an induc* *ed simplicial F2-algebra structure. Choose " 2 Spec(ß0A). Choosing a weak homoto* *py factorization, (R00, m) ! A00, of A at ", which exists by Proposition 1.1, then* *, by Propo- sition 1.1, [5, (3.8.3) & (3.10)], and [19, x5], (R00, m) is a Cohen-Macaulay r* *ing of depth zero and, hence, locally Artin. Thus, by Proposition 1.1, we may simply assume * *that R is locally Artinian, that A is a cofibrant simplicial commutative R -algebra,* * and that the unit map R ! ß0A is a surjective local homomorphism. We will now show that * *such A is Andr'e nilpotent at ". Note that if ` = k(") then char(`) = 2 since R and * *A have characteristic 2. Let B = A LR` = A R `. Then (3.12) C(B) ~=C(A) R ` ~=C(A)=mC(A). Thus æ : C(A) ! C(B) is a surjection and keræ = mC(A). Since R is locally Artin* *ian, (3.13) ms = 0 s 0. Cf. [19, 2.3]. Let a, b 2 m and let x, y 2 C(A) of degrees 2. By Proposition * *3.1 (3) and a straightforward induction, r r 2r r flr2(ax + by) = a2 fl2(x) + b fl2(y) modulo decomposables. Thus, by (3.13) and Proposition 3.1 (4), flr2(ax + by) = 0 for r 0. Hence, by* * a further induction, fl2 acts locally nilpotently on mC(A). Therefore, by Corollary 3.3, * *B is Andr'e nilpotent. We conclude, by the validity of the Nilpotence Conjecture at the pr* *ime 2, that A is a homotopy 2-intersection at ". * * 2 References [1]M. Andr'e, Homologie des alg`ebres commutatives, Die Grundlehren der Mathem* *atischen Wis- senschaften 206, Springer-Verlag, 1974. 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