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
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Department of Mathematics, Calvin College, 3201 Burton Street, S.E., Grand Ra*
*pids,
MI 49546
E-mail address: jturner@calvin.edu