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.
[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.
ON SIMPLICIAL ALGEBRAS WITH FINITE ANDR'E-QUILLEN HOMOLOGY 21
[5]L. L. Avramov, H.-B. Foxby, and B. Herzog, "Structure of local homomorphism*
*s," J. Algebra 164
(1994), 124-145.
[6]L. Avramov and S. Iyengar, Ä ndr'e-Quillen homology of algebra retracts,Ä *
*nn. Sci. Ecole Norm.
Sup. (4) 36 (2003), 431-462.
[7]L. Avramov and S. Iyengar, öH mological criteria for regular homomorphisms *
*and for locally com-
plete intersection homomorphisms,Ä lgebra, Arithmetic and Geometry (Mumbai*
*, 2000), T.I.F.R.
Studies in Math. 16 vol. I, Narosa, New Delhi, 2002; pp. 97-122.
[8]A.K. Bousfield, Ö perations on derived functors of non-additive functors," *
*manuscript, Brandeis
University 1967.
[9]H. Cartan, Ä lg`ebres d'Eilenberg-MacLane et homotopie," Expos'es 2 `a 11, *
*S'em. H. Cartan,
'Ec. Normale Sup. (1954-1955), Sect'etariat Math., Paris, 1956; [reprinted *
*in:] OEvres, vol. III,
Springer, Berlin, 1979; pp. 1309-1394.
[10]A. Dold, öH mology of symmetric products and other functors of complexes,Ä *
* nn. of Math. (2)
68 (1958), 40-80.
[11]W. Dwyer, öH motopy operations for simplicial commutative algebras," Trans.*
* A.M.S. 260 (1980),
421-435.
[12]P. Goerss, On the Andr'e-Quillen cohomology of commutative F2-algebras, Ast*
*'erique 186 (1990).
[13]P. Goerss and T. Lada, "Relations among homotopy operations for simplicial *
*commutative alge-
bras," Proc. A.M.S. 123 (1995), 2637-2641.
[14]J. Grodal, "The transcendence degree of the mod p cohomology of finite Post*
*nikov systems,"
Stable and unstable homotopy (Toronto, ON, 1996), Fields Inst. Commun., 19,*
* Amer. Math. Soc.,
Providence, RI, 1998, 111 - 130.
[15]T.H. Gulliksen, Ä homological characterization of local complete intersect*
*ions," Composition
math. 23, (1971), 251-255.
[16]J. Lannes and L. Schwartz, Ä propos de conjectures de Serre et Sullivan," *
*Invent. Math. 83 (1986),
593 - 603.
[17]S. Mac Lane, Homology, Springer-Verlag, New York (1967).
[18]__________, Categories for the Working Mathematician, Graduate Texts in Mat*
*hematics 5,
Springer-Verlag, Berlin, 1971.
[19]H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.
[20]H. Miller, "The Sullivan conjecture on maps from classifying spaces,Ä nnal*
*s of Math. 120 (1984),
39-87.
[21]D. Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer-*
*Verlag, 1967.
[22]__________, Ö n the (co)homology of commutative rings," Proc. Symp. Pure Ma*
*th. 17 (1970),
65-87.
[23]__________, Ö n the homology of commutative rings," Mimeographed Notes, M.I*
*.T.
[24]__________, äR tional homotopy theory,Ä nnals of Math. 90 (1968), 205-295.
[25]J.-P. Serre, öC homologie modulo 2 des espaces d'Eilenberg-MacLane," Commen*
*t. Math. Helv. 27
(1953), 198-231
[26]J. M. Turner, "Simplicial Commutative Fp-Algebras Through the Looking Glass*
* of Fp-Homotopy
Theory", in Homotopy Invariant Algebraic Structures, Contemporary Mathemati*
*cs, Meyer et. al.
editors (1999)
[27]__________, Ö n simplicial commutative algebras with vanishing Andr'e-Quill*
*en homology," In-
vent. Math. 142 (3) (2000) 547-558.
[28]__________, Ö n simplicial commutative algebras with Noetherian homotopy," *
*J. Pure Appl.
Alg. 174 (2002) pp 207-220.
Department of Mathematics, Calvin College, 3201 Burton Street, S.E., Grand Ra*
*pids,
MI 49546
E-mail address: jturner@calvin.edu
__