NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL COMMUTATIVE ALGEBRAS JAMES M. TURNER Abstract.In this paper, we continue a study of simplicial commutative al* *gebras with finite Andr'e-Quillen homology, that was begun in [19]. Here we re* *strict our focus to simplicial algebras having characteristic 2. Our aim is to find* * a generalization of the main theorem in [19]. In particular, we replace the finiteness c* *ondition on homotopy with a weaker condition expressed in terms of nilpotency for th* *e action of the homotopy operations. Coupled with the finiteness assumption on Andr'* *e-Quillen homology, this nilpotency condition provides a way to bound the height a* *t which the homology vanishes. As a consequence, we establish a special case of an o* *pen conjecture of Quillen. Introduction Throughout this paper, unless otherwise stated, all rings and algebras are co* *mmuta- tive. Given a simplicial supplemented `-algebra A, with ` a field having non-zero c* *harac- teristic, it was shown, in [19], that if its total Andr'e-Quillen homology HQ*(* *A) is finite (as a graded `-module) then its homotopy ß*A being finite as well implies that * *HQ*(A) is concentrated in degree 1. In this paper, we seek to find a generalization of* * this result by weakening the finiteness condition on homotopy. Thus we need to focus more o* *n its internal structure. As such we restrict our attention to the case where char` * *= 2 in order to take advantage the rich theory available in [10] and [11]. To be more specific, given such a simplicial algebra A having characteristic * *2, M. Andr'e [2] showed that the homotopy groups ß*A have the structure of a divided * *power algebra. Furthermore, W. Dwyer [10] showed that there are natural maps ffii: ßm A ! ßm+iA, 2 i m, which are homomorphisms for i < m and ffim = fl2, the divided square. All resu* *lting primary operations can now be described in terms of linear combinations of comp* *osites of the ffiis. Furthermore, there are Adem relations which allow any such compo* *site to be described in terms of admissible composites. This gives the set B of operat* *ions a non-commutative ring structure. ß*A then becomes an algebroid over B. Moreover,* * the module of indecomposables Qß*A inherits the structure of an unstable B-module. ___________ Date: February 11, 2002. 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 a grant from the National Science Foundation (USA). 1 2 JAMES M. TURNER A different perspective of Dwyer's operations can be taken in the following w* *ay. For 0 i n - 2 define ffi: ßnA ! ß2n-iA by ffi(x) = ffin-i(x). So, for example, ff0(x) = fl2(x). Written in this way, i* *teration of these reindexed Dwyer operations need not be nilpotent. Nevertheless, the main * *theorem of this paper shows a connection between the vanishing of Andr'e-Quillen homolo* *gy and the nilpotency of the Dwyer operations. Theorem A: Let ` be a field of characteristic 2 and let A be a connected simpl* *icial sup- plemented `-algebra such that the total Andr'e-Quillen homology HQ*(A) is finit* *e. Then a nilpotent action of ffn-2 on Qß*A implies that HQs(A) = 0 for s n. As a consequence, the following strengthens the main theorem of [19] and reso* *lves a conjecture posed in [20, 4.7] at the prime 2: Corollary: Let A be as in Theorem A and suppose that ß*A is locally nilpotent* * as a divided power algebra. Then HQs(A) = 0 for s 2. Note: The restriction to characteristic 2 is due to the need for an Adem relati* *ons among the homotopy operations ffii which insures that arbitrary composites can be wri* *tten in terms of admissible operations. At the prime 2, this was established in the wor* *k of Dwyer [10] and Goerss-Lada [12]. At odd primes, Bousfield's work [7] gives a prelimin* *ary version of Adem relations, but a final version still awaits to be produced. Connection to conjectures of Quillen. For a simplicial algebra A over a ring R * *the Andr'e-Quillen homology, D*(A|R; M), of A over R with coefficients in an A-modu* *le M was first defined by M. Andr'e and D. Quillen [1, 16, 17]. In particular, for a* * simplicial supplemented `-algebra A, we write HQ*(A) := D*(A|`; `). Next, recall that a homomorphism ' : R ! S of Noetherian rings is essentially* * of finite type if for each n 2 SpecS there is a factorization (0.1) R fi!R[X]N !ffSn where R[X] = R[X1, . .,.Xn] is a polynomial ring, N is a prime ideal in R[X] ly* *ing over n, the homomorphism ø : R ! R[X]N is the localization map, and the homomorphism* * oe is surjective. Furthermore, we call such a homomorphism a locally complete inte* *rsection if, for each n 2 SpecS, Ker(oe) is generated by a regular sequence. In [16, (5.6, 5.7)], Quillen formulated the following two conjectures on the * *vanishing of Andr'e-Quillen homology: Conjecture: Let ' : R ! S be a homomorphism essentially of finite type between Noetherian rings and assume further that Ds(S|R; -) = 0 for s 0. Then: I.Ds(S|R; -) = 0 for s 3; NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL ALGEBRAS 3 II.If, additionally, the flat dimension fdR S is finite, then ' is a locally * *complete intersection homomorphism. In [4], L. Avramov generalized the notion of local complete intersections to * *arbitrary homomorphisms of Noetherian rings. He further proved a generalization of Conjec* *ture II. to such homomorphisms. See P. Roberts review [18] of this paper for an exc* *ellent summary of these results and the history behind them. A proof of Conjecture II.* * was also given in [19] for homomorphisms with target S having non-zero characterist* *ic. We now indicate how Theorem A bears on providing a resolution to the above Co* *njec- ture. To formulate this, let R ! (S, `) be a homomorphism of local rings with c* *har` = 2. Let # : (Q TorR*(S, `))m ! (Q TorR*(S, `))2m-1 be the operation induced by ff1. We call this operation the Andr'e operation s* *ince it generalizes the operation studied by M. Andr'e [3] when S = ` and m = 3. Theorem B: Let ' : R ! (S, `) be a surjective homomorphism of local rings with char` = 2 and assume further that Ds(S|R; `) = 0 for s 0. Then 1. Ds(S|R; `) = 0 for s 3 if and only if the Andr'e operation # acts nilpot* *ently on Q TorR*(S, `); 2. ' is a complete intersection if and only if the divided square fl2 acts ni* *lpotently on Q TorR*(S, `). As an application of Theorem B, the following proves the vanishing portions o* *f the conjecture for certain homomorphisms in characteristic 2: Theorem C: Let R ! S be a homomorphism essentially of finite type between Noe- therian rings of characteristic 2 and assume further that Ds(S|R; -) = 0 for s * * 0. Then: 1. Ds(S|R; -) = 0 for s 3 provided R ! S is a homomorphism of Cohen-Macaulay rings; 2. If the flat dimension fdRS is finite, then ' is a locally complete interse* *ction. Note: L. Avramov showed in [4] that Conjecture I. holds when either R or S is a* * locally complete intersection. More recently, L. Avramov and S. Iyengar [5] have streng* *thened this special case of Conjecture I. by showing that it holds for homomorphisms R* * ! S for which there exists a composite Q ! R ! S which is a local complete intersec* *tion homomorphism. See [6] for a more leisurely discussion of their results. 4 JAMES M. TURNER Organization of this paper. The first section reviews the properties of the hom* *otopy and Andr'e-Quillen homology for simplicial commutative algebras and the methods* * for computing them, particularly in characteristic 2. The next section then focuse* *s on a device called the character map associated to simplicial algebras having finite* * Andr'e- Quillen homology. After showing that Dwyer's operations possess certain annihil* *ation properties, we show that the character map can be highly non-trivial. From this* * Theorem A easily follows. This enables us, in the third section, to prove Theorem B. Fi* *nally, the last section begins with establishing a chain level criterion for the nilpotenc* *y of Andr'e's operation. After a brief excursion into commutative algebra, we prove a special* * case of Theorem C and then show how the general case follows. Acknowledgements. The author would like to thank Lucho Avramov for sharing his expertise and insights and for a thorough and critical reading of an earlier dr* *aft of this paper. He would also like to thank Paul Goerss and Jean Lannes for helpful advi* *ce and discussions during work on this project. 1.Homotopy and homology of simplicial commutative algebras in characteristic 2 1.1. Operations on chains and homotopy. Again let A be a simplicial algebra of characteristic 2. We describe the algebra structure associated to A at two lev* *els: on the associated chain complex and on the associated homotopy groups. While only * *the latter will be needed in the process of proving Theorem A, the former descripti* *on will be needed in the subsequent applications. Let V be a simplicial vector space, over the field F2, and let C(V ) denote i* *ts associated chain complex. In [10], Dwyer constructs natural chain maps (1.2) k : (C(V ) C(V ))n+k ! C(V V )n for all 0 k n. They satisfy the relations (1.3) 0 + T 0T + OE0 = and (1.4) k + T kT + OEk = @ k-1+ k-1@. Here T : C(V ) C(V ) ! C(V ) C(V ) and T : C(V V ) ! C(V V ) are the tw* *ist maps. Also, OEk : C(V ) C(V ) ! C(V V ), k 0 is the degree -k map that is zero on [C(V ) C(V )]m for m 6= 2k, and, in degr* *ee 2k, is the projection on one factor: [C(V ) C(V )]2k= p+q=2kVp Vq ! Vk Vk. Finally : C(V ) C(V ) ! C(V V ) is the Eilenberg-Zilber map. Now given a simplicial F2-algebra (A, ~), define the maps (1.5) ffi: C(A)n ! C(A)2n-i for 0 i n - 1 NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL ALGEBRAS 5 by (1.6) x ! ~ i(x x) + ~ i-1(x @x). To describe the algebra structure on the associated chains of A, recall that * *a dg - algebra is a non-negatively differentially graded F2-algebra ( , @) together wi* *th maps flk : n ! kn for k 0 and n 2, satisfying the following relations 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 1 5. flk(xy) = xkflk(y) for x 2 0 and y 2 2 6. flk(fl2(x)) = fl2k(x) 7. @flk(x) = (@x)flk-1(x). Proposition 1.1. Let A be a simplicial F2-algebra. 1. The chain complex C(A) possesses a dg -algebra with ff0 = fl2; 2. For x 2 C(A)n and 0 < i < n - 1, @(ffi(x)) = ffi-1(@(x)), for 0 < i < n - 1, @(ff0(x)) = x . @(x), and @(ffn-1(x)) = ffn-2(@x) + x2. Proof: 1. See [2] and [11, x2 and 3]. 2. By 1.4, 1.6, and the Leibniz rule for @, @ffi(x) = ~@ i(x x) + ~@ i-1(x @x) = ~ i@(x x) + ~ i-1@(x @x) + ~ i(@x x) + ~ i(x @x) = ~ i-1(@x @x) = ffi-1(@x). The calculation of @ff0(x) and @ffn-1(x) from (1.3) and (1.4) is similar. * * 2 Note: For a divided power algebra of characteristic 2, it is enough to speci* *fy the action of divided square fl2 to determine all divided powers. Specifically, for* * x 2 flk = fls12(x) . fls22(x) . .f.lsr2(x), where k = 2s1+ . .+.2sr. Cf. [2] and [11, x2]. Now, the ffi induce the homotopy operations ffi: ßnA ! ß2n-iA, 0 i n - 2. Furthermore, Dwyer's higher divided squares ffii: ßnA ! ßn+iA, 2 i n are now defined as ffii[x] = [ffn-i(x)]. 6 JAMES M. TURNER In particular, ffin[x] = [ff0(x)] = fl2[x]. We now summarize the properties of the higher divided squares, as established* * in [10, 12] (see also [11, x2 and 3]). Proposition 1.2. The higher divided squares possess the following properties: 1. Adem relations: (a)For i < 2j, X ` j - i + s -'1 ffiiffij = ffii+j-sffis i+1_2 s i+j_3j - s (b)For j < i, X ` i - s -'1 ffiffj = ffi+2j-2sffs i+2j_3 s i+j-1_2s - j 2. Cartan formula: for x, y 2 ß*A, 8 >: 0 |x| > 0, |y| > 0. Let I = (i1, . .,.is) be a sequence of positive integers. Then call I admissi* *ble provided it 2it+1for all 1 t < s. Furthermore, define for I its excess to be the inte* *ger e(I) = (i1 - 2i2) + (i2 - 2i3) + . .+.(is-1- 2is) + is = i1 - i2 - . .-.* *is, its length to be the integer ~(I) = s, and its degree to be the integer d(I) = i1 + . .+.is. Also write ffiI = ffii1. .f.fiis ffI = ffi11. .f.fiss. As an application of the Adem relations, given any sequence I, ffiI can be wr* *itten as a sum of ffiJ's, with each J an admissible sequence. Similarly, by another app* *lication of the Adem relations, any ffI can be written as a sum of ffJ's, with J not nec* *essarily admissible. Finally, denote by B the algebra spanned by {ffiI|I admissible}. A B-module * *M is then called unstable provided ffiIx = 0 for any x 2 Mn, whenever I is admissibl* *e with e(I) > n. For example, given a simplicial supplemented `-algebra A then Qß*A, * *the module of indecomposables, is an unstable B-module. NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL ALGEBRAS 7 1.2. The homotopy of symmetric algebras. Let ` be a field having characteristic 2. We provide description of the homotopy groups of a very important type of si* *mplicial `-algebra, namely, the homotopy of S`(V ) - the symmetric algebra, or free comm* *utative algebra, generated by a simplicial vector space V over `. When ` = F2, we simpl* *y write S for S`. If W is a vector space, let K(W, n) denote the simplicial vector space with h* *omotopy ß*K(W, n) ~=W , concentrated in degree n. Then write S`(W, n) = S`(K(W, n)) S`(n) = S`(`, n) By a theorem of Dold [9], there is a functor of graded vector spaces S` and a* * natural isomorphism (1.7) ß*S`(V ) ~=S`(ß*V ). Again, we simply write S for S` when ` = F2. To describe the functor S` on graded vector spaces, we first note that it com* *mutes with colimits. Thus we need only describe S`(F`(n)) where F`(n) ~= `. We * *now recall the description of this functor when ` = F2. A proof of the following ca* *n be found, for example, in [10]: Proposition 1.3. S(F (n)) ~= [ffI(xn) : oe(I) < n - 1] ~= [ffiI(xn) : I admissible, e(I) < n]. Here [-] denotes the free divided power algebra functor. Note that QS(W ) is* * a free unstable B-module, for any positively graded vector space W . The functor S can be further decomposed as M S(-) = Sm (-). m 0 We review its description because of its importance below. First, define the weight of an element of S(W ) as follows: wt(u) = 1, wt(uv) = wt(u) + wt(v), wt(ffii(u)) = 2 wt(u) for u, v 2 W. Proposition 1.4. Let W be a graded vector space. Then Sm (W ) is the subspace S* *(W ) spanned by elements of weight m. Now, to describe S`, note first that, the uniqueness of adjoint funtors, ther* *e is a natural isomorphism of `-algebras j : S`(V F2`) ! S(V ) F2` where V is any F2-vector space. 8 JAMES M. TURNER Proposition 1.5. The natural isomorphism j in turn induces a natural isomorphism ~= j* : S`((-) F2`) ! S(-) F2` of functors to the category of -algebras. Moreover, for each i, m 2 and gra* *ded F2- vector space W , there is a commutative diagram ffii S`(W F2`)n !! S`(W F2`)n+i # # ffii F S(W )n F2` !! S(W )n+i F2` where F denotes the Frobenius map on `. Proof: Since ß*S`(-) ~=S`(-), by Dold's theorem, then the first point regardi* *ng j* follows from a Kunneth theorem argument. To prove the second part, it suffices * *to prove it for the graded vector space F`(n) = F (n) F2`, by a standard argument utili* *zing universal examples. In this case, the map j extends the map ` ! F2 F2` sending axn to xn a. By naturality of j* and the properties of Dwyer's operat* *ions, we have ffii(j*(axn))=ffii(xn a) = ffii((xn 1)(1 a)) = ffii(xn 1)(1 a)2 = (ffii(xn) 1)(1 a2) = ffii(xn) a2 which is the desired result. * * 2 It now follows that, for a simplicial `-vector space V , the generators and r* *elations for ß*S`(V ) are completely determined by Dwyer's result Proposition 1.2 and Do* *ld's theorem. 1.3. Andr'e-Quillen homology and the fundamental spectral sequence. We now provide a brief review of Andr'e-Quillen homology, for simplicial supplemented * *`-algebras, and the main computational device for relating homotopy and homology - Quillen's fundamental spectral sequence. Our primary source for this material is [11]. Cf* *. also [16, 17, 14]. Let A be a simplicial supplemented `-algebra. Then the Andr'e-Quillen homolog* *y of A is defined as the graded vector space HQ*(A) = ß*QX, where X is a cofibrant replacement of A, in the closed simplicial model structu* *re on simplicial supplemented `-algebras [14, 11]. Some standard properties of Andr'e-Quillen homology are summarized in the fol* *lowing: Proposition 1.6. [11, x4] Let A be a simplicial supplemented `-algebra. 1. If A = S`(V ), for some simplicial vector space V , then HQ*(A) ~=ß*V . NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL ALGEBRAS 9 f g 2. Let A ! B ! C be a cofibration sequence in the homotopy category of simp* *licial supplemented algebras. Then there is a long exact sequence HQ*(f)Q . .!.HQs+1(C) @!HQs(A) ! Hs (B) HQ*(g)Q @ Q ! Hs (C) ! Hs-1(C) ! . . . 3. If V is a vector space and [ , ] denotes morphisms in the homotopy categ* *ory of simplicial supplemented algebras, then the map [S`(V, n), A] ! Hom (V, IßnA) is an isomorphism. In particular, ßnA = [S`(n), A]. Another important tool we will need is the notion of connected envelopes for * *a simpli- cial supplemented algebra A. Cf. [19, x2]. These are defined as a sequence of c* *ofibrations j1 j2 jn jn+1 A = A(0) ! A(1) ! . .!. A(n) ! . . . with the following properties: (1) For each n 1, A(n) is a n-connected. (2) For s > n, HQsA(n) ~=HQsA. (3) There is a cofibration sequence fn jn S`(HQnA, n) ! A(n - 1) ! A(n). The following is proved in [19]: Lemma 1.7. If HQs(A) = 0 for s > n then A(n - 1) ~=S`(HQn(A), n) in the homot* *opy category of simplicial supplemented algebras. Finally, a very important tool for bridging the Andr'e-Quillen homology to th* *e homo- topy of a simplicial algebra is provided by the fundamental spectral sequence o* *f Quillen [16, 17]. For simplicial supplemented `-algebras in characteristic 2, we will n* *eed certain properties of this spectral sequence which can be described by combining the re* *sults of [11, x6] with Proposition 1.5. The following summarizes those features that we * *will need. Proposition 1.8. Let A be a simplicial supplemented `-algebra. Then there is a * *spectral sequence of algebras E1s,tA = Ss(HQ*(A))t F2` =) ßtA with the following properties: 1. For ß0A ~=`, E1s,tA = 0 for s > t and, hence, the spectral sequence conver* *ges; 2. The differentials act as dr : Ers,t! Ers+r,t-1; 3. The Dwyer operations ffii: Ers,t! Er2s,t+i, 2 i t have indeterminacy 2r-1 and satisfy the following properties: (a)Up to determinacy, the Adem relations and Cartan formula holds; 10 JAMES M. TURNER (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, r 2, are induced by the operations on Er-1A. T* *he operations on E1 A are induced by the operations on ErA with r < 1; and (d)The operations on E1 A are also induced by the operations on ß*A. Recall that, if Bqs,t Ers,t, q r is the vector space of elements that survive to Eqs,tbut have zero residue clas* *s in Eqs,t, then y 2 Ers,tis defined up to indeterminacy q if y is a coset representitive f* *or a particular element in Ers,t=Bqs,t. 2. Proof of Theorem A 2.1. The character map for simplicial algebras with finite homology. Fix a connected simplicial supplemented `-algebra A with HQ*(A) finite as a graded `-* *module. Define the Andr'e-Quillen dimension of A [6] to be AQ-dim A = max {s : HQs(A) 6= 0} and define the connectivity of A to be conn A = min{s : HQs(A) 6= 0} - 1. We assume that AQ-dim A 2. Let n = AQ-dim A. Then the (n-1)-connected envelope A(n - 1) has the property that A(n - 1) ~=S`(HQn(A), n) in the homotopy category. Cf. Lemma 1.7. Thus we have a map A ! S`(HQn(A), n) in the homotopy category with the property that it is an HQn-isomorphism. We now define the character map of A to be the resulting induced map of unsta* *ble B-modules (2.8) A : Qß*A ! Qß*S`(HQn(A), n) The importance of the character map is established by the following: Theorem 2.1. Let A be a connected simplicial supplemented `-algebra having fi* *nite Andr'e-Quillen dimension n. Then the character map A is non-trivial. Furthermo* *re, y 2 Qß*A can be chosen so that A(y) = ffsn-2(x), for some non-trivial x 2 HQn(A) and some s > 0. Thus fft-2acts non-nilpotently * *on y, for all 2 t n. NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL ALGEBRAS 11 As an immediate consequence, we are now in a position to supply the following: Proof of Theorem A: This follows immediately from Theorem 2.1 since if ffn-2 ac* *ts nilpotently on any x 2 Qß*A, the same must also hold for (x). Hence it follows* * that n > AQ-dim A. 2 2.2. Annihilation properties among some homotopy operations. Before we prove Theorem 2.1, we need to pin down specific annihilators of elements of B. To thi* *s end, define, for s, t 0, the operation `(s, t) = ffi2s+tffi2s+t-1. .f.fi2t+1. Proposition 2.2. Let J be a finite subset of {j|j > 2t} and let X , = ajffijwj j2J be a linear combination of admissible operations, with each aj 2 `. Then `(s, t* *), = 0 for s 0. Proof: It is sufficient to prove the result for , = ffij with j > 2t. Write j =* * 2t+ n with n 1. Note first that an application of the Adem relations shows that, for any* * t, ffi2t+1ffi2t+1= ffi2t+1ffi2t+2= 0. We thus assume, by induction, that, for each t and 0 < i < n, there exists s * *0 such that `(s, t)ffi2t+i= 0. By another application of the Adem relations, 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 0 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 12 JAMES M. TURNER 2.3. Proof of Theorem 2.1. As we noted previously, it is sufficient to show tha* *t, for some t > 0, fftn-2(x) = ffi2tffi2t-1. .f.fi2(x) 2 E12tsurvives non-trivially to* * E1 for some x 2 HQn(A), where n = AQ-dim A. Such an element will map non-trivially under * *with the desired properties, by Proposition 1.3. The strategy is to examine the induced map of spectral sequences {ErA} ! {ErS(HQn(A), n)} which is split surjective at E1. The goal is to show that the image of the spli* *tting on the indecomposables contains a non-trivial infinite cycle with the requisite specif* *ications. Now, assume n 2. Then the result holds when n = m and n = m + 1, where m = connA + 1, because in these cases Quillen's spectral sequence collapses [19* *] and, hence, A is a split surjection. Thus we can now induct on n - m. This further * *reduces to an induction on dim`HQm(A), which is finite by assumption. By the Hurewicz theorem [11, (8.3)], ßm A ~=HQm(A). By Proposition 1.6.3, a c* *hoice of a basis element y 2 HQm(A) is represented by a map oe : S(m) ! A of simplici* *al supplemented `-algebras, by Proposition 1.6.3. Let B be the homotopy cofibre (* *aka mapping cone [11, (4.5)]) of oe and let f : A ! B be the induced map. Note that* * there is an identity: (2.9) A = B (Qf*) Then dim`HQm(B) = dim`HQm(A) - 1. By induction, we assume that, for x 2 HQn(A), `(b, 0)x 2 E12b,tB survives non-trivially to E1 B and determines an element y0 * *2 ßtB such that (2.10) B (y0) = ffbn-2(x). Now, in the spectral sequence for A, Proposition 1.8.3 (b) tells us, by induc* *tion, that `(b, 0)x survives to some Er2b,tA with r 2b. Furthermore, by Proposition 1.4* *, the differential satisfies: ( [uy + ,y] r = 2a - 2b, a > b; dr[`(b, 0)x]= [uy] otherwise, with u 2 E12b+r-1and , a linear combination of admissible Dwyer operations. Our* * goal is to show that there exists s b so that `(s, 0)x is an infinite cycle. We ex* *amine the above cases on r in reverse order. r 6= 2a - 2b :We use induction on t - 2b - r, by first noting that for t - 2b -* * r = 0, dr[`(b, 0)x]2 Er2b+r,t-1= 0. Since `(b+1, 0) = ffi2b+1`(b, 0) 6= 0 then ffi2b+1* *`(b, 0)x survives to E2rby Proposition 1.8.3 (b). By Proposition 1.8.1 and Proposition 1.8.3 (a) * *and (d), d2r[ffi2b+1`(b, 0)x]= ffi2b+1[dr`(b, 0)x]= 0. Hence ffi2b+1`(b, 0)x survives to E2r+12b+1,t+2b+1. By assumption, we have (t + 2b+1) - 2b+1- (2r + 1) t + 2b+ r - 2b+1- 2r - 1 < t - 2b- r. Thus, by induction, `(s, 0)x is an infinite cycle for some s b + 1. NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL ALGEBRAS 13 P r = 2a - 2b :Write , = aIffiI as a homogeneous linear combination of admissi* *ble operations, with each aI 2 `. Then a typical indexing I can be written as I = (* *i1, . .,.ia). Admissibility implies that i1 2i2 . .2.a-1ia 2a > 2b. Proposition 2.2 now applies toetell us that `(s, b), = 0 for some s 0. Thus* * `(s, 0)x = `(s, b)`(b, 0)x survives to E2 r, where e = s - b, and er d2er[`(s, b)`(b, 0)x]= `(s, b)[dr`(b, 0)x]= 0 2 E2 . er+1 Thus `(s, 0)x = `(s, b)`(b, 0)x survives to E2 , so proceed as per the previo* *us case. Let y 2 ß*A be the element determined by [`(s, 0)x] 2 E1 A. To see that it is* * non- trivial, suppose that [`(s, 0)x]2 ErA is a boundary. Then Er(f)([`(s, 0)x]) = [* *`(s, 0)x] is also a boundary in ErB. But, since s b, this contradicts the induction hyp* *othesis. We conclude, by induction and Equation 2.10, that A(y) = B (ffs-bn-2(y0)) = ffs-bn-2 B (y0) = ffs-bn-2ffb(x) = ffsn* *-2(x). 2 3. Proof of Theorem B Theorem 3.1. Let R ! S ! ` be a surjective homomorphisms of Noetherian rings, with ` a field of characteristic 2, such that Ds(S|R; `) = 0 for s 0. Then th* *e following hold: 1. If the divided square fl2 acts nilpotently on Q TorR*(S, `) it follows tha* *t D 2(S|R; `) = 0. 2. If the Andr'e operation # acts nilpotently on Q TorR*(S, `) it follows tha* *t D 3(S|R; `) = 0. Proof: Using the simplicial model structure for simplicial commutative algebras* * [15, xII], let R ,! 2. (b)If S is Cohen-Macaulay then S0 is Artin and depthR0= 0. (c)R and S are both Cohen-Macaulay if and only if R0and S0 are both Artin * *local rings. Proof: For 1., see [8, 2.1.3 and 2.1.8]. For 2., see [8, 2.1.9]. For 3., form a surjection S ! (S0, n0) by quotienting out by the ideal genera* *ted by the maximal regular sequence in a minimal generating set for n. Let I R be t* *he kernel of R ! S ! S0. Form a surjection R ! (R0, m0) by, again, quotienting out* * by an ideal generated by the maximal regular sequence in a minimal generating set * *for I. Then there is a resulting commuting diagram for which the vertical maps are com* *plete intersections (i.e the kernels are generated by regular sequences). Now, applying the Jacobi-Zariski sequence [1, V.1] to the diagram above, we g* *et two long exact sequences . .!.Ds(S|R; -) ! Ds(S0|R; -) ! Ds(S0|S; -) ! Ds-1(S|R; -) ! . . . and . .!.Ds(R0|R; -) ! Ds(S0|R; -) ! Ds(S0|R0; -) ! Ds-1(R0|R; -) ! . . . From [1, VI.26], the long exact sequences reduce to two injections Ds(S|R; -) ! Ds(S0|R; -) and Ds(S0|R; -) ! Ds(S0|R0; -) for s 2. Furthermore, the first map is an isomorphism in the same range, as i* *s the second map for s > 2. Composing the two gives the desired map. Next, by 1., S is a Cohen-Macaulay ring if and only if S0 is Cohen-Macaulay. * *Since depth S0= 0 [8, 1.2] we have dim S0= 0, which occurs if and only if S0 is Artin* *ian. Cf. [13, x5]. Thus n0is nilpotent as an ideal. Now, if x 2 m0then OE0(x) 2 n0satisfies OE0(xt) = OE0(x)t= 0 for t 0. Thus xt2 KerOE0for t 0. From the construction, depth(Ker OE0) = 0 so we can c* *hoose y 6= 0 in Ker OE0 such that yxt = 0 for t 0. Choose the smallest t 1 such * *that yxt= 0. Then u = yxt-16= 0 satisfies ux = 0. We conclude that depthR0= 0. 16 JAMES M. TURNER Finally, if both R and S are both Cohen-Macaulay then we can conclude, from 3* *. (b), that S0 is Artin and that dim R0= 0, hence R0is Artin. The converse follows fro* *m [8, 2.1.3]. 2 4.3. Proof of Theorem C. We are now in a position to prove Theorem C. We first give a result which connects the Artinian property on a local ring to the nilpo* *tency of the Andr'e operation on Tor. Theorem 4.3. Let (R, m, `) ! S be a surjective homomorphism of local rings of* * char- acteristic 2. Then R being Artin implies that the Andr'e operation # acts nilpo* *tently on Q TorR*(S, `). Proof: Factor R ! S as per (4.13). Let w 2 mC 2(<). Then w = t1x1 + . .+.tnxn with t1, . .,.tn 2 m, x1, . .,.xn 2 C 2(<). From the properties of divided squares, we have s s 2s s fls2(w) = t21fl2(x1) + . .t.nfl2(xn) modulo decomposables . Since R is an Artin local ring, then ms = 0 for s 0, by [13, 2.3], hence fls2(w) = 0 modulo decomposables, s 0. Since fl2 kills decomposables in positive degrees, we conclude that fl2 acts ni* *lpotently on mC 2(<). The result now follows from Lemma 4.1. 2 Proof of Theorem C: By [1, S.29], it is enough to show that D 3(S|R; `) = 0, fo* *r any residue field ` = Sn=nSn, n 2 SpecS. By the stability of Andr'e-Quillen homolog* *y under localization [1, V.27] we may assume that ' : R ! (S, n, `) is a homomorphism o* *f local rings. Further, since we are assuming that ' is essentially of finite type, th* *ere is a factorization R fi!R[X]N = T !ffS. as per (0.1). Since ø is faithfully flat, flat base change [1, IV.54] tells us * *that Ds(T |R; `) ~=Ds(`[X]|`; `) = 0 for s 1. Applying the Jacobi-Zariski sequence [1, V.1], we conclude that Ds(S|R; `) ~=Ds(S|T ; `) for s 2. Note further that fdTS = fdRS, by a base change spectral sequence argument, and* * if R is Cohen-Macaulay, then T is also Cohen-Macaulay, by Lemma 4.2.1 and 4.2.2. T* *hus we may assume that ' = oe. Now suppose R and S are both Cohen-Maclaulay. Then Lemma 4.2.3 allows us to assume that R is an Artin local ring. Thus 1. follows from Theorem 4.3 and Theo* *rem B. Finally, if fdRS is finite, then Q TorR*(S, `) is finite and, hence, possesse* *s a nilpotent action of fl2. Thus ' is a complete intersection, by Theorem B, giving us 2. * * 2 NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL ALGEBRAS 17 References [1]M. Andr'e, Homologie des alg`ebres commutatives, Die Grundlehren der Mathem* *atischen Wis- senschaften 206, Springer-Verlag, 1974. [2]__________, üP issances divisees des algebres simpliciales en caracteristiq* *ue deux et series de Poincare de certains anneaux locaux," Manuscripta Math. 18 (1976), 83-108. [3]__________, äL (2p+1)-`eme d'eviation d'un anneau local," Enseignement Mat* *h. (2) 23 (1977), 239-248. [4]L. Avramov, öL cally complete intersection homomorphisms and a conjecture o* *f Quillen on the vanishing of cotangent homology,Ä nnals of Math. (2) 150 (1999), 455-487. [5]L. Avramov and S. Iyengar, Ä ndr'e-Quillen homology of algebra retracts," p* *reprint, Purdue Uni- versity (2001). [6]L. Avramov and S. Iyengar, öH mological criteria for regular homomorphisms * *and for locally complete intersection homomorphisms," preprint, Purdue University (2000). [7]A.K. Bousfield, Ö perations on derived functors of non-additive functors," * *manuscript, Brandeis University 1967. [8]W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, 1* *998 (revised). [9]A. Dold, öH mology of symmetric products and other functors of complexes,Ä * * nn. of Math. (2) 68 (1958), 40-80. [10]W. Dwyer, öH motopy operations for simplicial commutative algebras," Trans.* * A.M.S. 260 (1980), 421-435. [11]P. Goerss, On the Andr'e-Quillen cohomology of commutative F2-algebras, Ast* *'erique 186 (1990). [12]P. Goerss and T. Lada, "Relations among homotopy operations for simplicial * *commutative alge- bras," Proc. A.M.S. 123 (1995), 2637-2641. [13]H. Matsumura, Commutative ring theory, Cambridge University Press, 1986. [14]H. Miller, "The Sullivan conjecture on maps from classifying spaces,Ä nnal* *s of Math. 120 (1984), 39-87. [15]D. Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer-* *Verlag, 1967. [16]__________, Ö n the (co)homology of commutative rings," Proc. Symp. Pure Ma* *th. 17 (1970), 65-87. [17]__________, Ö n the homology of commutative rings," Mimeographed Notes, M.I* *.T. [18]P. Roberts, Review of öL cally complete intersection homomorphisms and a co* *njecture of Quillen on the vanishing of cotangent homology," Math. Reviews 2001a:13024. [19]J. M. Turner, Ö n simplicial commutative algebras with vanishing Andr'e-Qui* *llen homology," In- vent. Math. 142 (3) (2000) 547-558. [20]__________, "Simplicial Commutative Fp-Algebras Through the Looking Glass o* *f Fp-Homotopy Theory", in Homotopy Invariant Algebraic Structures, Contemporary Mathemati* *cs, Meyer et. al. editors (1999) Department of Mathematics, Calvin College, 3201 Burton Street, S.E., Grand Ra* *pids, MI 49546 E-mail address: jturner@calvin.edu