ON SIMPLICIAL COMMUTATIVE ALGEBRAS WITH NOETHERIAN
HOMOTOPY
JAMES M. TURNER
Abstract.In this paper, we introduce a strategy for studying simplicial *
*commuta-
tive algebras over general commutative rings R. Given such a simplicial *
*algebra A, this
strategy involves replacing A with a connected simplicial commutative k(*
*")-algebra
A("), for each " 2 Spec(ß0A), which we call the connected component of A*
* at ".
These components retain most of the Andr'e-Quillen homology of A when th*
*e coeffi-
cients are k(")-modules (k(") = residue field of " in ß0A). Thus these c*
*omponents
should carry quite a bit of the homotopy theoretic information for A. Ou*
*r aim will be
to apply this strategy to those simplicial algebras which possess Noethe*
*rian homo-
topy. This allows us to have sophisticated techniques from commutative a*
*lgebra at
our disposal. One consequence of our efforts will be to resolve a more g*
*eneral form of
a conjecture of Quillen that was posed in [13].
Overview
Our focus, in this paper, is to take the view that the study of Noetherian ri*
*ngs
and algebras through homological methods is a special case of the study of simp*
*licial
commutative algebras having Noetherian homotopy type. Our goal is to show that *
*such
simplicial algebras can be given a suitably rigid structure in the homotopy cat*
*egory,
which then allows us to bring in methods from commutative algebra. Such methods
should enable more facile techniques from homological algebra to be ferried in *
*for the
purpose of elaborating the global structure of such simplicial algebras.
To begin, we define for a simplicial commutative algebra A to have Noetherian*
* homo-
topy provided:
1. ß0A is a Noetherian ring, and
2. each ßm A is a finite ß0A-module.
If, more strongly, ß*A is a finite graded ß0A-module, we that A has finite No*
*etherian
homotopy.
In order to achieve a more systematic study of simplicial algebras with Noeth*
*erian
homotopy, particularly to allow us a straighter path to proving our main result*
*, Theorem
B below, we first seek to rigidify the action of ß0 from the homotopy groups to*
* the
simplicial algebra. This is accomplished by the following:
___________
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, Noetherian
homotopy.
Research was partially supported by a grant from the National Science Foundat*
*ion (USA).
1
2 JAMES M. TURNER
Theorem A: Any simplicial commutative algebra A is weakly equivalent to a conn*
*ected
simplicial supplemented ß0A-algebra.
Theorem A provides the means to import in methods from commutative algebra, m*
*ost
notably localizations and completions. In particular, we use these methods as a*
* means
to provide a proof of a conjecture posed in [13] which generalizes a conjecture*
* of Quillen
regarding the vanishing of Andr'e-Quillen homology. Our larger interests lie in*
* providing
an understanding of the of the homotopy type of a simplicial commutative algebr*
*a A
with Noetherian homotopy over a Noetherian ring R through its Andr'e-Quillen ho*
*mology
D(A|R; -). Here we shall view this homology as a functor of ß0A-modules. This e*
*nables
us to be specific about the homology's rigidity properties.
Before stating our result, we first need a homotopy invariant notion of compl*
*ete in-
tersection. To obtain one, we first define a map A ! B of simplicial commutati*
*ve
R-algebras, augmented over a field `, to be virtually acyclic provided D 1(B|A;*
* `) = 0.
Also, if W is a graded `-module, define the simplicial `-algebra So(W ) by
O
So(W ) = S(Wn, n)
n
where S(V, n) is the free commutative `-algebra generated by the Eilenberg-MacL*
*ane
space K(V, n).
Define a simplicial commutative R-algebra A over ` to be a homotopy n-interse*
*ction,
for n 1, provided there is a commutative diagram
R -! R0
j # # j0
A -! A0
# #
` -=! `
with the horizontal maps being virtually acyclic over ` and in the homotopy cat*
*egory
there is an isomorphism
A0 LR0` ~=So(W )
with W a graded `-module satisfying W>n = 0. We call a general simplicial commu*
*ta-
tive R-algebra A a locally homotopy n-intersection if, for each " 2 Spec(ß0A), *
*A is a
homotopy n-intersection over the residue field k(")
Recall that the flat dimension of an R-module M to be the positive integer fd*
*RM
such that
(0.1) fdRM m () TorRi(M, -) = 0 for i > m.
Theorem B: Let A be a simplicial commutative R-algebra with finite Noetherian h*
*o-
motopy, char(ß0A) 6= 0, and fdR(ß*A) is finite. Then Ds(A|R; -) = 0 for s 0 i*
*f and
only if A is a locally homotopy 1-intersection.
This resolves a conjecture posed in [13] generalizing a conjecture of Quillen*
* [11, 5.7].
ON SIMPLICIAL ALGEBRAS WITH NOETHERIAN HOMOTOPY 3
Notes:
1. Theorem B fails when char(ß0A) = 0, as shown in [13].
2. Theorem B fails for general simplicial algebras having Noetherian homotopy*
*. The
case of the simplicial algebras S(V, n) over a field of non-zero character*
*istic provide
counterexamples, by computations of Cartan [5].
3. A homomorphism between Noetherian rings is a locally complete intersection*
* if
and only if it is a locally homotopy 1-intersection, as shown in [2, 13].
Quillen further conjectured a more general result [11, 5.6] which drops the f*
*inite flat
dimension condition. We would like to indicate a possible simplicial version o*
*f this
conjecture of Quillen. To formulate it, we first indicate a special vanishing *
*result for
Andr'e-Quillen homology that we will prove.
Theorem C: Let A be a simplicial commutative R-algebra with Noetherian homotop*
*y.
Then Ds(A|R; -) = 0 for s 3 if and only if A is a locally homotopy 2-intersec*
*tion.
This now leads us to pose the following:
Conjecture: Let A have finite Noetherian homotopy with char(ß0A) 6= 0. Then
Ds(A|R; -) = 0 for s 0 implies that A is a locally homotopy 2-intersection.
The strategy for proving Theorem B is to show that Ds(A|R; k(")) = 0 for s *
*2 for
each " 2 Spec(ß0A). This is sufficient by a result of Andr'e [1, S.30]. Followi*
*ng a strat-
egy of Avramov [2], we use Theorem A coupled with commutative algebra techniques
developed in [3] to replace A with A("), its connected component at ", which ha*
*s the
following properties:
1. A(") is a connected simplicial supplemented k(")-algebra;
2. fdR(ß*A) < 1 implies that A(") has finite Noetherian homotopy;
3. Ds(A|R; k(")) ~=Ds(A(")|k("); k(")) for s 2.
Theorem B now follows from the algebraic version of a theorem of Serre establis*
*hed in
[13].
Acknowledgements. The author wishes to thank Lucho Avramov for sharing his ex-
pertise on commutative algebra and to Paul Goerss for sharing his expertise on *
*Postnikov
systems.
1. Postnikov Systems and Theorem A
Throughout this paper, we fix a commutative ring with unit and let Alg be *
*the
category of (unitary) commutative rings augmented over . Finally, we denote by*
* Alg
the category of -algebras in Alg .
We will also be assuming the reader has an acquaintance with closed (simplici*
*al) model
category theory. Our main resource is [10]. We will further need specific resul*
*ts on the
model category structure for simplicial commutative rings and algebras. Our pri*
*mary
sources are [10, 12, 6].
4 JAMES M. TURNER
1.1. Postnikov Systems. Let A be an object in the category sAlg of simplicial *
*com-
mutative rings over . We review the construction of a Postnikov tower for A de*
*rived
from [4, 7] which we will be use in the proof of Theorem A.
Following [7, x5], define the nth Postnikov section of A as follows: for fix*
*ed k, let
In,k! Ak be the kernel of the map
Y
d : Ak ! An
ffi:[m]![k]
where OE runs over all injectionsQin the ordinal number category with m n, d *
*is induced
by the maps OE* : Ak ! Am , and denotes the product in the category of algeb*
*ras
augmented over . Define
(1.2) A(n)k = Ak=In,k
Notice that there is a quotient map in sAlg , A ! A(n), and that if k n, A(n*
*)k = Ak.
There are also quotient maps
(1.3) qn : A(n) ! A(n - 1)
and A ~=limA(n). Let F (n) be the fibre of qn, i.e.
(1.4) F (n) = ker(qn : A(n) ! A(n - 1)).
qn
Note that F (n) ! A(n) ! A(n - 1) forgets to a fibration sequence as simplicial*
* abelian
groups. As such, the following can be proved just as in [7, 5.5].
Lemma 1.1. The homotopy groups of F (n) are computed as follows:
(
ßnA k = n;
ßkF (n) =
0 k 6= n.
1.2. Eilenberg-MacLane objects. Following [4, x5], define an object A of sAlg *
*to
be of type K if ß0A ~= and the higher homotopy groups of A are trivial. Suppo*
*se M
is a -module. We say that a map A ! B is of type K (M, n) n 1, if A is of ty*
*pe
K , ß0B ~= , ßnB ~=M (as a -module), all other homotopy groups of B are trivia*
*l,
and the map A ! B is a ß0-isomorphism.
For a general map f : A ! B in sAlg , let C be the pushout of the diagram B0*
*
A0! A(0)0obtained by using a functorial construction to replace A by a cofibran*
*t object
and the two maps A ! B and A ! A(0) by cofibrations. There is then a commutative
diagram
f
A - ! B
~" "~
f0 0
(1.5) A0 - ! B
# #
n(f)
A(0)0 - ! C(n + 1)
ON SIMPLICIAL ALGEBRAS WITH NOETHERIAN HOMOTOPY 5
The bottom map n(f) is called the difference construction of f. The following *
*can be
proved just as in [4, 6.3].
Proposition 1.2. Suppose that A ! B is a map of simplicial commutative algebras
which is a ß0-isomorphism and whose homotopy fibre F is (n-1)-connected. Let M*
* =
ßnF . Then M is naturally a -module for = ß0B and n(f) is a map of type
K (M, n + 1). If ßkF vanishes except for k = n, then the right-hand square in 1*
*.5 is a
homotopy fibre square.
1.3. Differentials functor. For an object A in Alg , define its -differentials*
* to be
the -module
D A = J=J2 A
where J is the kernel of the product A A ! A. As a functor to the category of
-modules, D posseses a right adjoint - the functor
(-)+ : Mod ! Alg
defined by M+ = M with the usual twisted product
(x, a) . (y, b) = (bx + ay, ab).
An equivalent identification of the differentials functor
(1.6) D ~=I=I2 A ,
where I is the augmentation ideal of A, which can be seen to follow from Yoneda*
*'s
lemma.
The next proposition is proved in [10, xII.5].
Proposition 1.3. The prolonged adjoint pair of functors
D : sAlg () sMod : (-)+
induces an adjoint pair on the homotopy categories
LD : Ho(sAlg ) () Ho(sMod ) : R(-)+.
Finally, the following useful property of the derived functor of differential*
*s follows
from [12, 7.3].
Proposition 1.4. If f : A ! B is a ß n-isomorphism, then LD (f) is a ß n-isomor*
*phism.
1.4. Characterizing K (M, n)-type. Fix a -module M. In sMod , the fibration pn*
* :
E(M, n) ! K(M, n) is determined by the Dold-Kan correspondence by to correspond*
* to
the map of normalized chain complexes {M !1 M} ! {M} with the source concentrat*
*ed
in degrees n and n-1, the target concentrated in degree n, and the map being th*
*e identity
in degree n and trivial otherwise.
Applying (-)+ to pn gives a K (M, n)-type fibration in sAlg
(pn)+ : E (M, n) ! K (M, n)
which we call the canonical map of type K (M, n).
6 JAMES M. TURNER
Proposition 1.5. Let A ! B be of type K (M, n) between cofibrant objects in sAl*
*g .
Then there is a commuting diagram in sAlg
A -~! E (M, n)
# # pn
B -~! K (M, n)
with the horizontal maps being weak equivalences.
Proof. To begin, note that the canonical map B ! is (n-1)-connected. Thus t*
*he
induced map D B ! 0 is (n-1)-connected by Proposition 1.4. Let I = ker(B ! ).
Filtering B by powers of I we note that B cofibrant implies that
Iq=Iq+1 = Sq(I=I2) ~=Sq(D B)
where the last identity always holds when the augmentation is surjective, by (1*
*.6). Thus
there is a convergent spectral sequence
E1p,q= Hp+q[Sq(D B)] =) ßp+qB.
From the connectivity indicated above and [12, 7.40], E1p,q= 0 for 0 < p+q 2(*
*q-2)+n.
Thus we obtain
M ~=ßnB ~=ßnD B.
Thus there is an n-connected map D B ! K(M, n) and its adjoint B ! K (M, n) will
be a weak equivalence by the computations above and the assumption that A ! B i*
*s of
type K (M, n).
Finally, A ! is a weak equivalence, hence D A ! 0 is a weak equivalence by
Proposition 1.4. Since A, and hence D A, are cofibrant, the composite D A ! D B*
* !
K(M, n) lifts to a map D A ! E(M, n), whose adjoint A ! E (M, n) is necessarily*
* a
weak equivalence. 2
1.5. Proof of Theorem A. Fix an object A in sAlg . We will show, by induction,
that there is a map X ! Y in s Alg and a commutative diagram in Ho(sAlg )
A(n) -~! X
(1.7) qn # #
A(n - 1) -~! Y
with the horizontal maps being equivalences. It is clear for n = 0 as A(0) ! *
*is a weak
equivalence.
Using 1.5, some closed model category theory and induction, we may assume that
there is a trivial fibration oe : A(n - 1)0 ! Y with the target Y a cofibrant*
* object in
s Alg .
Lemma 1.6. Let M = ßnA. Then there is a commuting diagram in Ho (sAlg ) of t*
*he
form
A(n - 1)0 - ! C(n + 1)
~# oe #~
Y - ! K (M, n + 1)
ON SIMPLICIAL ALGEBRAS WITH NOETHERIAN HOMOTOPY 7
with the top arrow from 1.5.
Proof. First, note that since oe : A(n - 1)0! Y is a trivial fibration between *
*suitably
cofibrant objects (see above) it follows from that and from 1.6 that
D oe : D A(n - 1)0! D Y
is a trivial fibration between cofibrant objects in sMod . By [10, I.1.7], D o*
*e has a
homotopy left inverse i (i O D oe ' IdD A(n-1)).
Next, utilizing Lemma 1.5, let t : A(n - 1)0 ! K (M, n + 1) be the composite *
*of
A(n - 1)0! C(n + 1) ! K (M, n + 1). Let w : D Y ! K(M, n + 1) be the composite
(D t) O i. Then w O D oe ' D t and the result now follows from Proposition 1.3.*
* 2
From the previous lemma, we may form the homotopy pullback diagram in s Alg
X -! E (M, n + 1)
(1.8) # # (pn)+
Y -! K (M, n + 1).
By Proposition 1.2, the diagram below is also a homotopy pullback in sAlg
A(n)0 - ! A(0)0
(1.9) q0n# # [qn]
A(n - 1)0 - ! C(n + 1).
By Proposition 1.5 and Lemma 1.6, there is an induced map of diagrams 1.9 to 1.8
in the category Ho (sAlg ). Since fibrations and pullbacks in sAlg are fibrat*
*ions and
pullbacks as simplicial groups, a computation of homotopy groups can be perform*
*ed
utilizing Lemma 1.1 to show that the induced map A(n)0! X is a weak equivalence.
This completes the induction step.
2.Andr'e-Quillen homology and Theorems B and C
2.1. Base change property of Andr'e-Quillen homology. Recall that the cotangent
complex of a simplicial R-algebra A is defined to be the object of Ho(ModA)
(2.10) L(A|R) := P|R P A
where the T -module T|S= J=J2, J = ker(T ST ! T ), denotes the Kahler differe*
*ntials
of an S-algebra T , and P ! A is a cofibrant replacement of A as a simplicial R*
*-algebra.
Note: As in x1.3, T|Sis left adjoint to the functor M 7! M T where the image*
* has
a T -algebra structure with M2 = 0.
Also recall that given another simplicial R-algebra B, the derived tensor pro*
*duct of A
and B to be the object of Ho(sModR)
A LRB := P R Q
where Q ! B is a cofibrant replacement of B.
We now derive a base change property for the cotangent complex following [12].
8 JAMES M. TURNER
Lemma 2.1. If TorRq(Ak, Bk) = 0 for all k 0 and all q > 0 then A LRB ' A *
*R B.
Proof. This follows immediately from the spectral sequence [10, xII.6]
E2p,q= ßpTorRq(A, B) =) ßp+q(A LRB).
2
Lemma 2.2. A RB|B ~= A|R R B
Proof. Let A0= A R B and fix an A0-module M. Then
hom A0( A0|B, M) ~=hom BAlgA0(A0, M A0)
~=hom RAlgA(A, M A)
~=hom A( A|R, M)
~=hom A0( A|R R B, M).
The result now follows from Yoneda's lemma. 2
Proposition 2.3. L(A LRB|B) ' L(A|R) LRB
Proof. Fix cofibrant replacements P and Q for A and B, respectively. Then
(2.11) L(A LRB|B) = P RQ|Q ~= P|R R Q
by Lemma 2.2. Since P is projective as a simplicial R-module then P|R is a pro*
*jective
P -module. Thus, by Lemma 2.1, the map P|R ~! P|R P A is a weak equivalence.
Since Q is projective, Lemma 2.1 further tells us that
(2.12) P|R R Q ~!( P|R P A) R Q ~=L(A|R) LRB
is a weak equivalence. The result now follows by combining 2.11 with 2.12. *
* 2
Corollary 2.4. As a functor of A R B-modules, D*(A LRB|B; -) ~=D*(A|R; -).
Proof. This follows from Proposition 2.3 and the identity D*(T |S; M) :=
ß*[L(T |S) T M]. 2
2.2. Proof of Theorem B. We first recall the main result of [13].
Theorem 2.5. Let A be a homotopy connected simplicial supplemented commutative
algebra over a field ` of non-zero characteristic. Then Ds(A|`; `) = 0 for s *
*0 implies
that there is an equivalence S`(D1(A|`; `), 1) ~=A in the homotopy category.
We now begin by establishing a special case of Theorem A. To that end let A b*
*e a
simplicial commutative R-algebra and assume that the unit R ! ß0A = is a surj*
*ection.
For " 2 Spec , define the connected component of A at " to be the connected sim*
*plicial
supplemented k(")-algebra
A(") = A LRk(").
Lemma 2.6. Let A be as above. Then
1. D*(A|R; k(")) ~=D*(A(")|k("); k(")), and
ON SIMPLICIAL ALGEBRAS WITH NOETHERIAN HOMOTOPY 9
2. if A also has finite Noetherian homotopy and fdR(ß*A) < 1 it follows that *
*A(")
has finite Noetherian homotopy.
Proof. 1. follows from Corollary 2.4. For 2., [10, xII.6] gives a spectral sequ*
*ence
E2s,t= TorRs(ßtA, k(")) =) ßs+t(A LRk(")).
From the finiteness conditions, each E2s,tis a finite k(")-module and vanishes *
*for s, t 0.
Thus A LRk(")) has finite Noetherian homotopy. 2
Corollary 2.7. Let A be as in Lemma 2.6.2 and further assume that char(k(")) 6=*
* 0.
Then Ds(A|R; k(")) = 0 for s 0 implies that Ds(A|R; k(")) = 0 for s 2.
Proof. This follows from Lemma 2.6 and Theorem 2.5. 2
Now assume that the simplicial algebra A in question is a homotopy connected *
*simpli-
cial supplemented -algebra, by Theorem A. We further assume that A has Noether*
*ian
homotopy.
Fix " 2 Spec and let d(-)denote the completion functor on R-modules at ". De*
*fine
the homotopy connected simplicial supplemented b-algebra A0by
A0= A L b.
Proposition 2.8. Suppose A is a simplicial commutative R-algebra, with R a Noet*
*her-
ian ring. Then ß*A0 ~=dß*Aand there exists a (complete) Noetherian R0 that fit*
*s into
the following commutative diagram in Ho(sRAlg )
''
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. j0 induces a surjection j0*: R0! ß0A0;
4. fdR(ß*A) finite implies that fdR0(ß*A0) is finite
Proof: First, Quillen's spectral sequence [10, II.6] Tor*(ß*A, b) =) ß*A0 colla*
*pses to
give the first result since b is flat over and each ßm A is finite over [9,*
* 8.7 and 8.8].
ffi0 ''0*
Next, by [3, 1.1], the unit ring homomorphism R ! b factors as R ! R ! b wi*
*th
OE having the properties described in 1. and j0*is a surjection. Thus the induc*
*ed map
j0: R0! A0induces a surjection on ß0, giving 3., and the desired diagram commut*
*es.
Now, by the transitivity sequence [12, 4.12] applied to R ! A ! A0, 2. follow*
*s from
the isomorphism
D*(A0|A; k(")) ~=D*(b | ; k(")) ~=0
which follows from Corollary 2.4.
Finally, 4. follows from [3, 3.2], as A has Noetherian homotopy. *
* 2
10 JAMES M. TURNER
Now, let A have finite Noetherian homotopy with Ds(A|R; -) = 0 for s 0. From
Proposition 2.8, Theorem 2.5, Corollary 2.7, and [1, xS.30], if fdR(ß*A) < 1 th*
*en
A(") ~=Sk(")(D1(A|R; k("), 1), for each " 2 Spec(ß0A), if and only if D(A|R; -)*
* = 0.
Thus Theorem B follows from the definition of locally homotopy complete interse*
*ction
(see introduction) and a transitivity sequence argument.
2.3. Proof of Theorem C. Let A be a simplicial commutative R-algebra with Noe-
therian homotopy. It follows from Lemma 2.6.1, Proposition 2.8, and [1, xS.30]*
*, that
D 3(A|R; -) = 0 if and only if D 3(A(")|k("); k(")) = 0, for all " 2 Spec(ß0A).*
* From
the definition of locally virtual homotopy complete intersection (see introduct*
*ion), Theo-
rem C will follow if we can show that, for each prime ideal ", A(") ~=So(D 2(A|*
*R; k(")))
in the homotopy category. But this in turn follows from [13, (2.2)].
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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