ON SIMPLICIAL COMMUTATIVE ALGEBRAS WITH VANISHING
ANDR'E-QUILLEN HOMOLOGY
JAMES M. TURNER
Abstract.In this paper, we study the Andr'e-Quillen homology of simplici*
*al commu-
tative `-algebras, ` a field, having certain vanishing properties. When *
*` has non-zero
characteristic, we obtain an algebraic version of a theorem of J.-P. Ser*
*re and Y. Umeda
that characterizes such simplicial algebras having bounded homotopy grou*
*ps. We fur-
ther discuss how this theorem fails in the rational case and, as an appl*
*ication, indicate
how the algebraic Serre theorem can be used to resolve a conjecture of D*
*. Quillen for
algebras of finite type over Noetherian rings, which have non-zero chara*
*cteristic.
Overview
Algebraic Serre Theorem. The following topological theorem is due to J.-P. Serre
[17] at the prime 2 and to Y. Umeda [19] at odd primes.
Serre's Theorem. Let X be a nilpotent space such that Hs(X; Fp) = 0 for s 0 a*
*nd
each Hs(X; Fp) is finite dimensional. Then the following are equivalent
1. ßs(X) Z=p = 0, s 0;
2. ßs(X) Z=p = 0, s 2.
In [1, 15, 16], M. Andr'e and D. Quillen constructed the notion of a homology*
* D*(A|R; M)
for a homomorphism R ! A of simplicial commutative rings, with coefficients in *
*a sim-
plicial A-module M. These homology groups can be defined as ß*(L(A|R) A M) whe*
*re
the simplicial A-module L(A|R) is called the cotangent complex of A over R.
We now propose an algebraic analogue of Serre's Theorem for simplicial augmen*
*ted `-
algebras. To accomplish this we will take simplicial homotopy ß*(-) to be the a*
*nalogue
of H*(-; Fp) and HQ*(-) = D*(-|`; `) to be the analogue of ß*(-) Z=p.
Algebraic Serre Theorem. Let A be a homotopy connected (i.e. ß0A = `) simplicial
supplemented commutative `-algebra, with char` 6= 0, such that ß*A is a finite *
*graded
`-module. Then the following are equivalent
1. HQs(A) = 0, s 0;
2. HQs(A) = 0, s 2.
___________
Date: February 11, 2002.
1991 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Second*
*ary: 13D40.
Key words and phrases. simplicial commutative algebras, Andr'e-Quillen homolo*
*gy, local complete
intersections, connected envelopes, Poincar'e series.
Research was partially supported by an NSF-NATO postdoctoral fellowship and b*
*y an NSF grant.
1
2 JAMES M. TURNER
We shall prove this theorem by following Serre's original approach in [17]. T*
*his will
require pooling technical tools such as an analogue of the notion of connected *
*covers
of spaces and various ways for making computations of the homotopy and homology*
* of
simplicial commutative algebras.
The algebraic Serre theorem cannot hold in general when the ground field has *
*charac-
teristic zero. At the end of x2, we indicate a partial result in the rational c*
*ase and point
to some examples that show that a full version of our theorem cannot hold ratio*
*nally.
Connections to Quillen's Conjecture. D. Quillen has conjectured that the cotan-
gent complex has certain rigidity properties. In particular, we recall the foll*
*owing, which
can be found in [15, (5.7)]:
Quillen's Conjecture. If A is an algebra of finite type over a Noetherian ring *
*R, such
that A has finite flat dimension over R and fdAL(A|R) is finite, then A is a qu*
*otient of
a polynomial ring by an ideal generated by a regular sequence.
Earlier results of Lichtenbaum-Schlessinger [9], Quillen [15], and Andr'e [1]*
* prove that
an R-algebra A is a complete intersection if and only if fdAL(A|R) 1. In char*
*acteristic
0 the conjecture was proved by Avramov-Halperin [3]. The general case was prove*
*d by
L. Avramov. Furthermore, Avramov characterized those homomorphisms R ! A of
Noetherian rings having locally finite flat dimension with fdAL(A|R) < 1. See [*
*2] for
details.
As a consequence of the Algebraic Serre Theorem, we have the following:
Theorem 0.1. Quillen's conjecture holds provided the algebra A has non-zero c*
*harac-
teristic.
Proof. Since fdAL(A|R) N if and only if Ds(A|R; -) = 0 for s > N then we se*
*ek
to show that the latter implies Ds(A|R; -) = 0 for s 2. By [1, (S.30)], it is*
* enough
to show that, for each prime ideal " A, Ds(A|R; k(")) = 0 for s 2, where k(*
*") is
the residue field of A". Since A has non-zero characteristic then each k(") has*
* prime
characteristic. Let ` denote a fixed residue field.
Since A is an algebra of finite type over R, then the unit map factors as R !*
* R[X] ff!
A, with X a finite set and oe a surjection. Since R ! R[X] is a flat homomorphi*
*sm, then
Ds(R[X]|R; `) ~=Ds(`[X]|`; `) = 0 for s 1, by [1, (4.54, 6.26)]. An applicati*
*on of [1,
(5.1)] now implies that Ds(A|R; `) ~=Ds(A|R[X]; `) for s 2. Since fdRA = fdR[*
*X]A,
by a change-of-rings spectral sequence argument, we may thus assume that R ! A *
*is
surjective.
Let F be the homotopy pushout over ` of R ! A in the simplicial model category
of simplicial commutative R-algebras over ` (see [14, 15, 16, 8] for general di*
*scussions
pertaining to this model structure). Then F is a connected simplicial suppleme*
*nted
commutative `-algebra with the properties
D*(F|`; `) ~=D*(A|R; `)
and
SIMPLICIAL ALGEBRAS WITH VANISHING ANDR'E-QUILLEN HOMOLOGY 3
ß*F ~=TorR*(A, `),
the first isomorphism following from the flat base change property for Andr'e-Q*
*uillen
homology [16, (4.7)] while the second follows from an argument utilizing the Ku*
*nneth
spectral sequence of Theorem 6.b in [14, xII.6].
By the assumption that fdRA < 1, it follows that ß*F is a finite graded `-mod*
*ule.
The result now follows. 2
Generalizing Quillen's Conjecture. We propose the following simplicial generali*
*za-
tion of Quillen's conjecture.
Conjecture. Let R be a Noetherian ring and let A be a simplicial commutative R-
algebra with the following properties:
(1) ß0A is a Noetherian ring having non-zero characteristic;
(2) ß*A is finite graded as a ß0A-module;
(3) fdRß*A < 1.
Then Ds(A|R; -) = 0 for s 0 implies Ds(A|R; -) = 0 for s 2.
Note. The condition on the characteristic of R is clearly needed, as noted abov*
*e.
A proof of this conjecture can be given when stronger conditions on ß0A are a*
*ssumed.
See [18]. For example, by the same reduction to the algebraic Serre theorem per*
*formed
in the proof of Theorem 0.1, the following special case can be proved.
Theorem 0.2. The conjecture holds if property (1) is replaced by the stronger*
* property
(10)ß0A is an algebra of finite type over R.
Organization of this paper. In the first section, we review the needed notions *
*of the
model category structure of simplicial supplemented commutative algebras. In pa*
*rticu-
lar, we review the construction and some properties of the homotopy and Andr'e-*
*Quillen
homology for simplicial commutative algebras. In the next section, we introduc*
*e the
notion of n-connected envelopes for simplicial commutative algebras which duali*
*zes the
notion of n-connected covers of spaces. We then pause to record a crucial split*
*ting result
and discuss specific types of simplicial commutative algebras which demonstrate*
* the fail-
ure of the algebraic Serre theorem rationally. We then, in the third section, d*
*iscuss the
properties of the Poincar'e series for the homotopy of a simplicial commutative*
* algebra.
This leads to the last section where we give a proof of the algebraic Serre the*
*orem.
Acknowledgements. The author would like to thank Haynes Miller and Paul Goerss
for several conversations relating to this project, Jean Lannes for his generou*
*s hospi-
tality while the author was staying in France, as well as for discussing severa*
*l areas
related to this topic, and Lucho Avramov for enlightening the author on many as*
*pects
of commutative algebra and for reading and commenting on several drafts of this*
* paper.
The author would also like to thank the referee for helping to effectively stre*
*amline the
presentation contained here.
4 JAMES M. TURNER
During the time this and related projects were being worked on, the author ha*
*d been
a guest visitor at the I.H.E.S., the Ecole Polytechnique, and Purdue University*
*. Many
thanks to each of these institutions for their hospitality and the use of their*
* facilities.
1. The homotopy and homology of simplicial commutative algebras
We now review the closed simplicial model category structure for sA` the cate*
*gory
of simplicial commutative `-algebras augmented over `. We will assume the read*
*er is
familiar with the general theory of homotopical algebra given in [14].
We call a map f : A ! B in sA` a
(i)weak equivalence (!~) () ß*f is an isomorphism;
(ii)fibration (!!) () f provided the induced canonical map A ! B xi0B ß0A is a
surjection;
(iii)cofibration(,!) () f is a retract of an almost free map [7, p. 23].
Theorem 1.1. [14, 12, 7] With these definitions, sA` is a closed simplicial m*
*odel cate-
gory.
For a description of the simplicial structure, see section II.1 of [14]. The *
*details will
not be needed for our purposes. Given a simplicial vector space V , over a fiel*
*d `, define
its normalized chain complex NV by
(1.1) NnV = Vn=(Im s0 + . .+.Imsn)
P n
and @ : NnV ! Nn-1V is @ = i=0(-1)idi. The homotopy groups ß*V of V is defined
as
ßnV = Hn(NV ), n 0.
Thus for A in sA` we define ß*A as above. The Eilenberg-Zilber theorem (see [1*
*0])
shows that the algebra structure on A induces an algebra structure on ß*A.
If we let V be the category of `-vector spaces, then there is an adjoint pair
S : V () A` : I,
where I is the augmentation ideal function and S is the symmetric algebra funct*
*or. For
an object V in V and n 0, let K(V, n) be the associated Eilenberg-MacLane obj*
*ect in
sV so that (
V s = n;
ßsK(V, n) =
0 s 6= n.
Let S(V, n) = S(K(V, n)), which is an object of sA` called a sphere algebra.
Now recall the following standard result which will be useful for us (see sec*
*tion II.4
of [14]).
Lemma 1.2. If V is a vector space, A a simplicial commutative algebra, and [*
* , ]
denotes morphisms in Ho(sA`), then the map
[S(V, n), A] ! Hom V(V, IßnA)
is an isomorphism. In particular, ßnA = [S(n), A], where S(n) = S(`, n).
SIMPLICIAL ALGEBRAS WITH VANISHING ANDR'E-QUILLEN HOMOLOGY 5
Here V is the category of vector spaces.
Thus the primary operational structure for the homotopy groups in sA`is deter*
*mined
by ß*S(Vo) for any Vo in sV. By Dold's theorem [6] there is a triple S on grade*
*d vector
spaces so that
(1.2) ß*S(V ) ~=S(ß*V )
encoding this structure. If char` = 0, S is the free skew symmetric functor an*
*d, for
char` > 0, S is a certain free divided power algebra (see, for example, [4, 7, *
*13]).
Recall [16] that given a map of simplicial commutative rings R ! S, there is a
functorially defined simplicial S-module S|R called the Kaehler differentials *
*of S
over R. Replacing S by a cofibrant simplicial R-algebra model X then the cotang*
*ent
complex of S over R is defined as the cofibrant simplicial S-module
L(S|R) := X|R X S
and the Andr'e-Quillen homology of S over R with coefficients in a simplicial S-
module M is defined as
D*(S|R; M) := ß*(L(S|R) S M).
For A in A`, define the indecomposable functor to be QA = I(A)=I2(A) which is*
* an
object of V. Define the homology functor HQ*(-) : sA` ! grV [7, 8, 12] by
HQs(A) = ßsQX, s 0,
where we choose a factorization
` ,! X ~!!A
of the unit ` ! A as a cofibration and a trivial fibration. This definition is *
*independent
of the choice of factorization as any two are homotopic over A (note that every*
* object
of sA` is fibrant). It is straightforward to show [7, (A.1)] that
B|` B ` ~=QB,
for any augmented `-algebra B, and so
HQ*(A) = D*(A|`; `).
We now summarize methods for computing homotopy and Andr'e-Quillen homology
that we will need for this paper.
Proposition 1.3. (1)If f : A !~ B is a weak equivalence in sA`, then HQ*(f) :
~=
HQ*(A) ! HQ*(B) is an isomorphism. The converse holds provided Iß0A = 0, t*
*hat
is, A is homotopy connected.
(2) There is a Hurewicz homomorphism h : Iß*A ! HQ*(A) such that if A is homot*
*opy
connected and HQs(A) = 0 for s < n then A is (n - 1)-connected and
~=
i.h : ßnA ! HQn(A) is an isomorphism and
~= Q
ii.h : ßn+1A ! Hn+1(A) is a surjection, which is also injective for n > *
*1.
6 JAMES M. TURNER
f g
(3) Let A ! B ! C be a cofibration sequence in Ho(sA`). Then: There is a l*
*ong
exact sequence
HQ*(f)Q
. .!.HQs+1(C) @!HQs(A) ! Hs (B)
HQ*(g)Q @ Q
! Hs (C) ! Hs-1(C) ! . . .
Proof. For all of these, see [7, IV]. In particular, (2) follows from Quillen*
*'s fundamental
spectral sequence and the connectivity of Dold's functor S [6]. *
* 2
2. Connected Envelopes
In this section, we construct and determine some properties of a useful tool *
*for study-
ing simplicial algebras.
Given A in sA`, which is homotopy connected, we define its connected envelopes
to be a sequence of cofibrations
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 existence of a connected envelopes is a consequence of the following :
Proposition 2.1. Let A in sA` be (n - 1)-connected for n 1. Then there exist*
*s a
map in sA`,
fn : S(HQnA, n) ! A,
with the following properties
1. fn is an isomorphism on ßn and HQn;
2. the homotopy cofibre M(fn) of fn : S(HQnA, n) ! A is n-connected and sati*
*sfies
HQsM(fn) ~=HQsA for s > n;
3. if HQsA = 0, s 6= n > 0 then fn is an isomorphism in Ho(sA`).
Proof. (1.) By the Hurewicz theorem, Proposition 1.3 (2), the map h : ßnA ! H*
*QnA
is an isomorphism. By Lemma 1.2 we have an isomorphism
[S(HQnA, n), A]~= Hom V(HQnA, IßnA).
Choosing fn to correspond to the inverse of h gives the result.
(2.) This follows from (1.) and the transitivity sequence
HQs+1M(fn) ! HQsS(HQnA, n) ! HQsA ! HQsM(fn).
SIMPLICIAL ALGEBRAS WITH VANISHING ANDR'E-QUILLEN HOMOLOGY 7
(3.) By (1.), fn : S(HQnA, n) ! A is an HQn-isomorphism and hence a weak equ*
*iva-
lence by Proposition 1.3(1). The converse follows from the computation
HQsS(V, n) = ßsQS(V, n) = ßsK(V, n) = V
for s = n and 0 otherwise. 2
Applications.
Proposition 2.2. If there is a cofibration sequence in sA`
S(V, n - 1) ! A ! S(W, n)
for some vector spaces V and W and some n > 1, then in Ho(sA`)
A ~=S(HQn-1A, n - 1) S(HQnA, n).
Proof. Proposition 1.3 (3) tells us that HQsA = 0 for s 6= n, n - 1, and the*
*re is an
exact sequence
0 ! HQnA ! V ! W ! HQn-1A ! 0.
Thus A is n - 2 connected and a connected envelope gives a cofibration
j Q
S(HQn-1A, n - 1) !iA ! S(Hn A, n)
for which HQ (j) is an isomorphism. Lemma 1.2 and Proposition 1.3 (2) give a co*
*mmu-
tative diagram
j* Q Q
[S(HQnA, n), A] -! [S(Hn A, n), S(Hn A, n)]
~=# #~=
j* Q Q
Hom (HQnA, ßnA) -! Hom (Hn A, ßnS(Hn A, n))
~=# h* h* #~=
~=
Hom (HQnA, HQnA) -! Hom (HQnA, HQnA)
which shows that j splits up to homotopy. 2
From Proposition 2.1 (3), if char` = 0 and V finite-dimensional then HQ*S(V, *
*n) ~=
V concentrated in degree n and ß*S(V, n) is free skew-commutative on a basis o*
*f V
concentrated in degree n. Thus ß*S(V, n) is bounded for any odd n, showing that*
* the
algebraic Serre theorem cannot hold rationally. On the other hand, we do have *
*the
following
Proposition 2.3. Let A be a connected simplicial augmented commutative `-algebr*
*a,
with char ` = 0, such that ß*A is a finite graded `-module. Then if HQodd= 0 a*
*nd
HQsA = 0 for s 0 we can conclude that Iß*A = 0.
Proof. 1. Suppose HQmA 6= 0 implies that 2r m 2s. Then HQmA(2r) 6= 0 impl*
*ies
that 2(r + 1) m 2s. Furthermore, HQoddA = 0 and ß*A(2r) is a finite graded*
* `-
module by a spectral sequence argument [14, xII.6] (using the fact that ß*S(V, *
*2r) is
8 JAMES M. TURNER
finitely-generated polynomial, when V is finite). The result follows by an indu*
*ction on
s - r, given that the result is certainly true for A < r, 1 >' S(2r), for any r*
*. 2
Example. Here is another example of a type of rational simplicial algebra wit*
*h finite
homotopy and Andr'e-Quillen homology.
Since ß*S(2r) ~=`[x2r], let f : S(2rs) ! S(2r) represent xs2r. Define A < r, *
*s > to be
the cofibre of f. Then the cofibration sequence extends to
S(2r) ! A < r, s >! S(2rs + 1).
The computation of ß*(A < r, s >) can be achieved by a Serre spectral sequence *
*ar-
gument (see the proof of Lemma 3.1) and the computation of HQ*(A < r, s >) can *
*be
obtained from Proposition 2.1 (3) using Proposition 1.3 (3). In the end, we obt*
*ain
(
` m = 2ri, 0 i < s,
ßm (A < r, s >) =
0 otherwise
and 8
><` m = 2r,
HQm(A < r, s >) = ` m = 2rs + 1,
>:
0 otherwise.
3.The Poincar'e Series of a Simplicial Algebra
Let A be a homotopy connected simplicial supplemented commutative `-algebra s*
*uch
that ß*A is of finite-type. We define its Poincar'e series by
X
#(A, t) = (dim `ßnA)tn.
n 0
If V is a finite-dimensional vector space and n > 0 we write
#(V, n, t) = #(S(V, n), t).
Combining the work of [5] withP[17, 19], this latterPseries converges in the op*
*en unit disc.
Given power series f(t) = aiti and g(t) = biti we define the relation f(t*
*) g(t)
provided ai bi for each i 0.
Lemma 3.1. Given a cofibration sequence
A ! B ! C
of connected objects in A` with finite-type homotopy groups, then
#(B, t) #(A, t)#(C, t)
which is an equality if the sequence is split.
SIMPLICIAL ALGEBRAS WITH VANISHING ANDR'E-QUILLEN HOMOLOGY 9
Proof. First, there is a Serre spectral sequence
E2s,t= ßs(C ßtA) =) ßs+tB.
This follows from Theorem 6(d) in xII.6 of [14], which gives a 1st-quadrant spe*
*ctral
sequence
E2*,*= ß*(B A ß*A) ) ß*B,
where ß*A is an A-module via the augmentation A ! ß0A. Here we can assume our
cofibration sequence is a cofibration with cofibre C. Since A is connected, the*
*n B A
ß*A ~=C ß*A.
Thus we have
X X
#(A, t)#(C, t) = ( dim`E2i,j)tn #(B, t).
n i+j=n
Finally, if the cofibration sequence is split then the spectral sequence collap*
*ses, giving
an equality. 2
Now given two power series f(t) and g(t) we say f(t) ~ g(t) provided limt!1f(*
*t)=g(t)
= 1. Given a Poincar'e series #(V, n, t), for a finite-dimensional `-vector sp*
*ace V and
n > 0, let
'(V, n, t) = logp#(V, n, 1 - p-t).
Then the following is a consequence of Th'eor`eme 9b in [17] and its generaliza*
*tion to
arbitrary non-zero characteristics in [19], utilizing the results of [5] to tra*
*nslate into our
present venue.
Proposition 3.2. For V an `-vector space of finite dimension q and n > 0 then '*
*(V, n, t)
converges on the real line and
'(V, n, t) ~ qtn-1=(n - 1)!.
4. Proof of the Algebraic Serre Theorem.
Recall that A is to be a connected simplicial augmented commutative `-algebra*
* with
HQ*(A) bounded and ß*A a finite graded `-module. The approach we take is to mim*
*ic
the proof of Serre's Theorem in [17]; utilizing higher connected envelopes, in *
*place of
higher connected covers, and Poincar'e series for homotopy, in place of Poincar*
*'e series for
homology. Unfortunately, owing to the nature of cofibration sequences, Serre's *
*original
proof runs into a glitch at the start in our situation. Fortunately, if we ski*
*p the first
step and evoke Proposition 2.2, the remainder of Serre's proof works without a *
*hitch.
Proof of the Algebraic Serre Theorem. Let
n = max {s|HQs(A) 6= 0}.
We must show that n = 1.
Consider the connected envelope
S(HQs(A), s) ! A(s - 1) ! A(s)
10 JAMES M. TURNER
for each s. From the theory of cofibration sequences (see section I.3 of [14]) *
*the above
sequence extends to a cofibration sequence
A(s - 1) ! A(s) ! S(HQs(A), s + 1).
Thus, by Lemma 3.1, we have
#(A(s), t) #(A(s - 1), t)#(HQs(A), s + 1, t).
Starting at s = n - 2 and iterating this relation, we arrive at the inequality
n-2Y
#(A(n - 2), t) #(A, t) #(HQs(A), s + 1, t).
s=1
Now, A(n - 1) ~=S(HQn(A), n) by Proposition 2.1 (3), but, by Proposition 2.2 (1*
*) and
Lemma 3.1, we have
#(A(n - 2), t) = #(HQn-1(A), n - 1, t)#(HQn(A), n, t).
Since ß*(A) is of finite-type and bounded then there exists a D > p such that #*
*(A, t)
D, in the open unit disc. Combining, we have
n-2Y
#(HQn-1(A), n - 1, t)#(HQn(A), n, t) D #(HQs(A), s + 1).
s=1
Applying a change of variables and logpto the above inequality, we get
n-2X
'(HQn-1(A), n - 1, t) + '(HQn(A), n, t) d + '(HQs(A), s + 1).
s=1
By Proposition 3.2, there is a polynomial f(t) of degree n - 2, a non-negative *
*integer a,
and positive integers b and d such that
atn-2 + btn-1 d + f(t), t 0
which is clearly false for n > 1. Thus n = 1. The rest of the proof follows f*
*rom
Proposition 2.1 (3). 2
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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