COMMUTATIVE ALGEBRA OF UNSTABLE K - MODULES,
LANNES' T - FUNCTOR AND
EQUIVARIANT MOD - P COHOMOLOGY
by
Hans-Werner Henn
Abstract
Let p be a fixed prime and let K be an unstable algebra over the mod - p Ste*
*enrod algebra A such
that K is finitely generated as graded Fp- algebra. Let Kfg-U denote the abe*
*lian category of finitely
generated K - modules with a compatible unstable A - module structure. We st*
*udy various concepts
of commutative algebra in this setting. The r^ole of the prime ideal spectru*
*m of a commutative ring
is here taken by a category R(K) which, roughly speaking, consists of the A *
*- invariant prime ideals
of K together with certain "Galois information"; sheafs will correspond to f*
*unctors on this category,
and the r^ole of the sheaf associated to a module will be taken by the compo*
*nents of Lannes' T -
functor. We discuss the notions of support, of a - torsion modules (for an i*
*nvariant ideal a of K)
and of localization away from the Serre subcategory T ors(a) of a - torsion *
*modules in our setting.
We show that the category Kfg- U has enough injectives and use these injecti*
*ves to study these
localizations and their derived functors; they are closely related to the de*
*rived functors of the a
- torsion functor Fa. Our results are formally analogous to Grothendieck's r*
*esults in the classical
situation of modules over a noetherian commutative ring R [Gr].
Important for applications is the case K = H*BG, the mod - p cohomology of a*
* classifying space
of a compact Lie group (or a suitable discrete group), and M = H*GX where X *
*is a (suitable) G -
CW - complex. In these cases the category R(K) and the functor on R(K) assoc*
*iated to H*GX can
be described in terms of group theoretic and geometric data, and our theory *
*yields a far-reaching
generalization of a result of Jackowski and McClure [JM] resp. of Dwyer and *
*Wilkerson [DW2]. As
a concrete application of our theory we describe the size of the kernel of t*
*he restriction map from the
unknown mod - 2 cohomology of the S - arithmetic group GL(n; Z[1=2]) to the *
*known cohomology
of its subgroup Dn of diagonal matrices.
0. Introduction
Let p be a prime number and K be an unstable algebra over the mod - p Steenrod *
*algebra A
[S, SE]. An unstable K - A - module is an unstable A - module together with an *
*A - linear
map K M -! M which defines on M the structure of a K - module. We will ususall*
*y drop
the A from the notation and say that M is unstable K - module. The mod - p coho*
*mology
H*X of a space X is the main source of examples for unstable algebras and examp*
*les for
unstable H*X - modules are given by the mod - p cohomology of spaces over X, or*
* more
generally of Thom spaces of vector bundles on spaces over X.
1
We will be mostly concerned with the case that K is finitely generated as an Fp*
* - algebra,
i.e. noetherian, and that M is finitely generated as K - module. We will call*
* such an M
an unstable finitely generated K - module and denote the abelian category of su*
*ch modules
by Kfg - U (cf. section 1). Interesting geometric examples are provided by th*
*e case of
equivariant mod - p cohomology (cf. Theorem 0.2). Our first result reads as fol*
*lows.
THEOREM 0.1. Let K be an unstable noetherian algebra. Then the category Kfg - U*
* has
enough injectives.
We will prove this in section 1 by actually constructing enough injectives. If *
*V is an elemen-
tary abelian p - group (i.e. V ~=(Z=p)n for some natural number n) and if K = H*
**BV , the
mod - p cohomology of the classifying space of V , such a theorem was proved by*
* Lannes and
Zarati [LZ2], and in fact, they even determined all indecomposable injectives. *
*We would like
to point out that there would not be enough injectives if we worked with module*
*s which are
finitely generated using both the K and A - module structure together, as the c*
*ase K = Fp
(i.e. the case of ordinary unstable A - modules) shows (cf. [LSc]).
Now assume that K is noetherian. We will use these injectives to study localiza*
*tion functors
on the category Kfg - U away from suitable subcategories. However, before we wi*
*ll get to
this we need to discuss various concepts of commutative algebra in Kfg - U.
The r^ole of the prime ideal spectrum in the classical case will be taken by an*
* appropriate
category R(K) (cf. 1.3 for a precise definition and section 2 for a discussion *
*of R(K)). Here
we just recall that the objects of R(K) are morphisms of unstable algebras ' : *
*K -! H*BV
where V is an elementary abelian p - group such that ' makes H*BV into a finite*
*ly generated
K - module. (Observe that Rad (Ker '), the radical of the kernel of ' is a prim*
*e ideal which
is invariant with respect to the action of the Steenrod reduced power operation*
*s and, in fact,
all such "invariant" prime ideals are obtained in this way (cf. 2.3).)
___
For each unstable finitely generated K - module M we have a functor M from R(K*
*) to
Kfg - U, which sends (V; ') to the "component" TV_(M; ') where T denotes Lannes*
*' functor
(cf. 1.4 for precise definitions). The functor M should be considered as the a*
*nalogue of the
sheaf fM which one associates to M in the classical case of modules over a comm*
*utative ring.
In fact, if one considers R(K)op as a site equipped with the trivial Grothendie*
*ck topology
then sheaves on this site are precisely covariant functors on R(K) (cf. [MM, II*
*I.2 and III.4]).
We will not go into the details of this analogy but we will point out further a*
*nalogies when
it seems appropriate. For a further justification for this philosophy we refer *
*to the comments
after Theorem 0.4. The discussion of support in section 2, in particular 2.10, *
*suggests also to
consider TV (M; ') as an unstable analogue of the appropriate stalk of fM, i.e.*
* as an analogue
of the classical localization of M at the prime ideal Rad (Ker '). For earlier *
*work on relations
between Lannes' functor and classical localization we refer to work by Dwyer an*
*d Wilkerson
[DW1].
For the remainder of the introduction we concentrate on the case of equivariant*
* mod - p
cohomology. This case is easier to explain and is also particularly interesting*
* because here
2
the category R(K) can be understood in group theoretic terms and the components*
* of the
T - functor are of geometric and group theoretic significance (cf. Theorem 0.2*
*). However,
we stress that the main results (Theorem 0.4 and Corollary 0.5 below) are true *
*for unsta-
ble finitely generated K - modules over an unstable noetherian algebra K (Theor*
*em 3.9
and Corollary 3.10) and the proof in the case of equivariant cohomology require*
*s the same
machinery as in the general case.
To state Theorem 0.2 we introduce some notation. As usual we denote the classif*
*ying space
of the topological group G by BG, the total space of the universal principal G *
*- bundle over
BG by EG and the mod - p cohomology of the Borel construction EG xG X by H*GX. *
*Fur-
thermore, for a fixed prime p, let A(G) denote the category whose objects are t*
*he elementary
abelian p - subgroups of G and whose morphism sets consist of those group homom*
*orphisms
which are induced by conjugation with an element in G [Q].
THEOREM 0.2. Fix a prime p. Assume we are in one of the following cases.
a) G is a compact Lie group and X is a G - CW - complex with finitely many G - *
*cells.
b) G is a discrete group for which there exists a mod - p acyclic G - CW - comp*
*lex F with
finitely many G - cells and finite isotropy groups, and let X be any G - CW - c*
*omplex, again
with finitely many G - cells and with finite isotropy groups (e.g. X = F ).
c) Let G be a profinite group such that the continuous mod - p cohomology H*cG *
*is finitely
generated as Fp - algebra.
I. Then H*BG is an unstable noetherian algebra and there is a canonical equival*
*ence of
categories
A(G) ! R(H*BG); E 7! (E; resG;E)
with resG;Edenoting the restriction homomorphism H*BG -! H*BE.
II. Furthermore H*GX is an unstable finitely generated H*BG - module and there *
*are iso-
morphisms
TE (H*GX; resG;E) ~=H*CG(E)(XE )
which are natural in E 2 A(G).
(Here CG (E) is the centralizer of E in G, XE denotes the E - fixed points of E*
* acting on X,
cohomology is with coefficient in Fp, and if G is profinite we assume that X is*
* a point and
we read H*GX as H*cG.)
If X = F (in case b)), then the spaces XE are also mod - p acyclic by Smith the*
*ory, and we
obtain the following easy but important consequence.
COROLLARY 0.3. Assume G satisfies the assumptions of Theorem 0.2.b). Then there*
* are
isomorphisms
TE (H*BG; resG:E) ~=H*BCG (E) ;
_
which are natural in E. |_|
3
Statement II of Theorem 0.2 and the Corollary are to a large extent due to Lann*
*es. Case (a)
is proved in the important and still unpublished preprint [L1] (cf. [L3] for a *
*proof in case G
is finite). The cases (b) and (c) are consequences of part (a) (for G finite); *
*(c) was proved
in [H2] and (b) will be proved in the appendix of this paper. In the case that *
*G is a group
of finite virtual cohomologial dimension (f.v.c.d.) the corollary was also prov*
*ed in [L1]. The
finite generation part of Statement I is well known in case (a); for part (b) w*
*e refer again to
the appendix. The equivalence of categories is essentially a folk result (cf. [*
*HLS2, I.5.3]).
Interesting classes of groups which are covered by (c) are p - adic analytic gr*
*oups in the
sense of Lazard [Lz], while (b) covers the case of (S) - arithmetic groups [Se]*
*, mapping class
groups of orientable surfaces [H], outer automorphism groups of free groups [CV*
*] (these are
all groups of f.v.c.d.) and word-hyperbolic groups in the sense of Gromov [GH].
Now we turn to localizations. We consider an invariant ideal a in H*BG, i.e. a *
*is invariant
with respect to the action of the Steenrod reduced power operations. Then the *
*class of
unstable finitely generated H*BG - modules which are annihilated by some power *
*of a (we
will call such modules unstable a - torsion modules) forms a Serre class and we*
* will study
localization away from the full subcategory T ors(a) of such modules.
These localizations away from T ors(a) are closely related to the right derived*
* functors RiFa
of the functor which associates to M its largest unstable a - torsion submodule*
* FaM. The
ideal a determines a subcategory O(a) of A(G), namely the full subcategory of a*
*ll objects E
for which a 6 Rad (Ker resG;E), i.e. for which a is not contained the radical o*
*f the kernel of
the restriction map resG;E: H*BG -! H*BE. The category O(a) should be thought o*
*f as
the analogue of the open complement (in the classical prime ideal spectrum) of *
*the subset
V (a) which is defined by a.
As usual we denote the inverse limit of a functor F defined on a category C by *
*limCF and
its derived functors by limiCF . Our main result reads now as follows.
THEOREM 0.4. Let G and X be as in Theorem 0.2 and let a be an invariant ideal i*
*n H*BG.
a) Then there is a natural exact sequence
0 -! FaH*GX -! H*GX -ae!limO(a)H*CG(E)(XE ) -! R1FaH*GX -! 0
in which the components of ae are induced by the obvious inclusions on the grou*
*p and space
level. In particular, the kernel and cokernel of ae are unstable finitely gene*
*rated a - torsion
modules. Furthermore, ae is localization away from the subcategory T ors(a).
b) There are natural isomorphisms
limiO(a)H*CG(E)(XE ) ~=Ri+1FaH*GX
for all i > 0. In particular, limiO(a)H*CG(E)(XE ) is an unstable finitely gene*
*rated a - torsion
module for all i > 0.
4
For the generalization of this result to the case where K is noetherian and M i*
*s in Kfg - U
the reader is referred to 3.9.
If one ignores the assertion about finite generation (which is a consequence of*
* Theorem 0.1)
this result is formally analogous to the classical situation [Gr]. There one co*
*nsiders an ideal
a in a noetherian commutative ring R and the derived functors of the functor a,*
* which
associates to an R - module M its a - torsion submodule. These derived funcors *
*are identified
with the cohomology groups H*V (a)(specR; fM) of spec(R) with support in the cl*
*osed set V (a)
and coefficients in the sheaf fM. The inverse limit and its derived functors co*
*rrespond in this
picture to the cohomology of the open complement of the closed set V (a) with c*
*oefficients in
Mf (The analogy can be made more precise by noting that O(a) is, in fact, an o*
*pen subobject
in the topos of sheafs on the site A(G)op [AGV,IV.8.4], and by refering to the *
*topos theoretic
versions of cohomology with support (cf. [AGV,V.6.5]).
In the important special case where a is the invariant maximal ideal m of all p*
*ositive dimen-
sional elements in H*BG, the submodule Fm is the largest unstable finite H*BG -*
* submodule
of M and we will write F instead of Fm . The category O(m) turns out to be the *
*full subcat-
egory of A(G) consisting of all non-trivial E. We will write A*(G) for this cat*
*egory. Then
Theorem 0.4 takes the following form.
COROLLARY 0.5. Let G and X be as in Theorem 0.2.
a) Then there is a natural exact sequence
0 -! F H*GX -! H*GX -ae!limA*(G)H*CG(E)(XE ) -! R1F H*GX -! 0
in which the components of ae are induced by the obvious inclusions on the grou*
*p and space
level. In particular, the kernel and cokernel of ae are finite. Furthermore, *
*ae is localization
away from the subcategory of unstable finite H*BG - modules.
b) There are natural isomorphisms
limiA*(G)H*CG(E)(XE ) ~=Ri+1F H*GX
for all i > 0. In particular, limiA*(G)H*CG(E)(XE ) is is finite for all i > 0.
If G is compact Lie and X is a point it is a theorem of Jackowski and McClure t*
*hat ae is an
isomorphism and all higher limits vanish in 0.5 [JM]. An algebraic version of t*
*he theorem of
Jackowski and McClure was proved by Dwyer and Wilkerson [DW2] and their result *
*motivated
the investigations in this paper. Applications in the case where ae is an isom*
*orphism (in
particular in the cases considered in [JM] and [DW2]) were discussed by Mislin *
*[M].
For applications of Theorem 0.4 and Corollary 0.5 the reader is referred to [H2*
*,H3,H4] where
we study the mod - p cohomology of groups of units in maximal orders of certain*
* p - adic
division algebras, of the general linear groups GL(p - 1; Zp) (with Zp the ring*
* of p - adic
integers) and the mod - 2 cohomology of SL(3; Z[1=2]) and GL(3; Z[1=2]). Appli*
*cations to
mapping classs groups will be considered in joint work with F. Cohen and Y. Xia.
5
The higher limits resp. the derived funcors of F in the case H*GX, G elementary*
* abelian will
be investigated in [HLO]. They yield new invariants for G - complexes which giv*
*e obstructions
for equivariant embeddings of finite G - complexes into smooth G - manifolds.
As another concrete application of our theory in the case of group cohomology w*
*e offer the
following result. Here Dn is the subgroup of diagonal matrices in the general *
*linear group
GL(n; Z[1=2]), cohomology is with coefficients in F2 and the size of a graded f*
*initely generated
module M over a connected finitely generatedPFp - algebra is given by the order*
* of the pole
at t = 1 of the power series series idimFpMiti. Note that the size measures t*
*he growth of
the sequence of numbers dimFp Mi.
THEOREM 0.6. The kernel of the restriction map aen from H*BGL(n; Z[1=2]) to H**
*BDn
has size precisely equal to n - n0 + 1 where n0 denotes the smallest natural nu*
*mber such that
aen0 is not a monomorphism and n n0. In particular, the size of the kernel of *
*aen0 itself is
1, i.e. Ker aen0 is periodic in large degrees.
By unpublished work of Dwyer it is known that n0 is finite but no element in th*
*ese kernels
seems to be known explicitly.
Here is a brief outline of the paper. In section 1 we review what we need to kn*
*ow about Lannes'
functor, discuss injectives and prove Theorem 0.1. Section 2 is concerned with *
*concepts of
commutative algebra in Kfg- U: we discuss invariant ideals and the category R(K*
*), torsion
modules, the T - support of an unstable finitely generated module over a noethe*
*rian algebra
K and its relation to the classical support. In section 3 we study localizatio*
*n away from
subcategories of torsion modules, we prove the generalization of Theorem 0.4 in*
* the context
of Kfg-U and discuss its consequences. We also show how to derive the theorem o*
*f Jackowski
and McClure resp. Dwyer and Wilkerson with our methods. Section 4 is devoted to*
* the proof
of Theorem 0.6 and in an appendix we prove part (b) of Theorem 0.2.
Acknowledgements:_______The research in this paper was inspired by the work of *
*Jackowski and
McClure [JM] and in particular by the algebraic approach to it by Dwyer and Wil*
*kerson
[DW2]. The paper should also be considered as a sequal to [HLS2]. In fact, my f*
*irst proof
(in 1990) of the fact that the kernel and cokernel of the map ae in Corollary 0*
*.5 (resp. 3.10)
were finite used the main result of [HLS2] in an essential way. I had helpful *
*discussions
with many different people on the subject matter of this paper and I am especia*
*lly happy to
acknowledge numerous inspiring discussions with Jean Lannes. In particular, he *
*first showed
me a result like Corollary 0.5 (resp. 3.10) in the case K = H*BV . I would also*
* like to thank
John Greenlees for a timely conversation on local cohomology and Bob Oliver for*
* comments
on a preliminary version of this paper. During the research presented in this *
*paper I was
supported by a Heisenberg fellowship of the DFG.
6
1. Review of Lannes' T - functor; injectives in Kfg - U
We begin by recalling some terminology and facts about Lannes' T - functor. As*
* general
reference for background information we refer to [L2,L3] and [S].
1.1______Let p be a fixed prime. Let U resp. K denote the category of unstabl*
*e modules resp.
unstable algebras over the mod - p Steenrod algebra A. The Steenrod algebra is *
*actually a
Hopf algebra and its diagonal gives rise to a tensor product on the categories *
*U resp. K.
For a fixed unstable algebra K we consider the following category K -U: its obj*
*ects, which we
call unstable K - A -modules (or unstable K - modules for short), are unstable *
*A - modules
M together with A - linear structure maps K M -! M which make M into a K - mod*
*ule;
its morphisms are all A - linear maps which are also K - linear. The full subc*
*ategory of
K - U consisting of those objects which are finitely generated as K - modules i*
*s denoted by
Kfg - U. Its objects will be called unstable finitely generated K - modules.
1.2.______Now let V be an elementary abelian p - group (i.e. V ~=(Z=p)n for som*
*e natural number
n). Let TV : U -! U be the functor introduced by Lannes [L2,L3]. It is left adj*
*oint to tensor-
ing with H*BV , so there are natural isomorphisms Hom U(TV M; N) ~=Hom U (M; H**
*BV N)
for all unstable modules M and N. TV has a number of remarkable properties. In *
*particular,
TV commutes with tensor products and lifts to a functor from K to itself and th*
*e adjunction
relation continues to hold in K: Hom K(TV K; L) ~= Hom K(K; H*BV L) for all u*
*nstable
algebras K and L. Similarly, TV lifts to a functor from K - U to TV K - U.
1.3.______To an unstable algebra K we associate a category S(K) as follows. Its*
* objects are the
morphisms of unstable algebras ' : K ! H*BV with V an elementary abelian p - *
*group;
it will be convenient to denote such an object as (V; '). Then the set of morp*
*hisms from
(V1; '1) to (V2; '2) are all homomorphisms V1 -ff!V2 of abelian groups such tha*
*t '1 = ff*'2.
The full subcategory of S(K) of objects (V; ') for which H*BV becomes a finitel*
*y generated
K - module via ' will be denoted by R(K). Note that in this case the homomorphi*
*sms ff
has to be injective. If K is a noetherian algebra then this category is equival*
*ent to a finite
category and it (resp. its opposite) was first investigated by Rector [R]. The *
*full subcategory
of R(K) having as objects all (V; ') with V non-trivial will play an important*
* role for us.
We will denote it by R*(K).
1.4.______Now consider the unstable algebra TV K. A morphism of unstable algebr*
*as ' : K -!
H*BV determines a connected component TV (K; ') of TV K: it is defined as TV (K*
*; ') :=
Fp(')T0VKTV K where Fp(') denotes Fp considered as a module over TV0K (the suba*
*lgebra of
homogeneous elements of degree 0) via the adjoint of '. Similarly, if M is in K*
* -U, we have a
"component" TV (M; ') := Fp(') T0VKTV M which has an obvious structure of an un*
*stable
TV (K; ') - module. Furthermore there is a canonical map of unstable algebras *
*aeK;(V;'):
K -! TV (K; ') which makes TV (M; ') into an unstable K - module; the map aeK;(*
*V;')is the
7
composition of the map flK;(V;'): K -! H*BV TV (K; '), which is adjoint to the *
*projection
map TV K -! TV (K; '), followed by the projection map H*BV TV (K; ') -! TV (K;*
* ').
Similarly, there are maps aeM;(V;'): M -! TV (M; ').
Now it is straightforward to check that the assignment (V; ') 7! TV (M; ') exte*
*nds to a
functor S(K) ! K - U. If K is noetherian and M 2 Kfg - U, then we obtain a fun*
*ctor
R(K) -! Kfg - U (cf. 1.8 and 1.12 below).
1.5.______Next we discuss injectives in the category K - U. First we have the *
*analogues of the
Brown - Gitler modules in the category U, i.e for each natural number n there i*
*s an unstable
K - module JK (n) representing the functor M 7! (Mn )*. In fact, if F (n) denot*
*es the free
unstable A - module on a generator in degree n, then we define JK (n)l, the sub*
*space of
elements of degree l, as ((K F (l))n)* with ( )* denoting the vector space dua*
*l. The A -
and K - module structure on JK (n) can then be defined by appropriate maps betw*
*een the
modules K F (.), just as in the case of the modules J(n) (cf. [LZ1]).
This description makes it clear that the module JK (n) is trivial in degrees bi*
*gger than n and
isLof finite type if K is of finite type. Furthermore, in U the module JK (n) i*
*s isomorphic to
* n-i *
iJ(i)(Kn-i) (with (K ) denoting the dual of the subspace of homogeneous *
*elements
of degree n - i and being considered as unstable module concentrated in degree *
*0).
To get more injectives we use the following refinement of the adjunction proper*
*ty of the T -
functor.
PROPOSITION 1.6 [LZ2]. Let K and L be two unstable algebras, V an elementary ab*
*elian p
- group and g : K -! H*BV L a homomorphism of unstable algebras. For any unstab*
*le K -
module M and unstable L - module N the adjunction Hom U(TV M; N) ~=Hom U (M; H**
*BV
N) induces an isomorphism
Hom TVK-U (TV M; N) ~=Hom K-U (M; H*BV N) :
(Here H*BV N is_a K - module via g and N is a TV K - module via eg: TV K -! L,*
* the
adjoint of g.) |_|
Now consider a map ' : K -! H*BV of unstable algebras and apply this propositi*
*on to
the map flK;(V;'): K -! H*BV TV (K; '). If N is an unstable TV (K; ') - modul*
*e, we
will also write H*BV (') N for the unstable K - module H*BV N if its K - modu*
*le
structure is defined via the map flK;(V;'). Because injectives in TV (K; ')-U a*
*re also injective
in TV K - U, exactness of TV [L2,L3] implies the following result.
PROPOSITION 1.7. Let ' : K - ! H*BV be a map of unstable algebras and_I be any
injective in TV (K; ') - U. Then H*BV (') I is injective in K - U. |_|
In particular, all the objects H*BV (') JTV(K;')(n) are injective. If K is und*
*erstood from
the context, and if (V; ') is in S(K), we will also write I(V;')(n) for this in*
*jective.
8
1.8.______We will be mainly concerned with the case of unstable noetherian alge*
*bras and finitely
generated unstable modules over them. We recall that in this case TV K is again*
* noetherian
and TV M is finitely generated over TV K ([DW1], [H1]). Furthermore, the canon*
*ical map
K -! TV K makes TV K into a finitely generated K - module and hence TV M become*
*s a
finitely generated K - module (cf. 1.12 below).
If K is also connected, then the map flM;(0:ffl): M -! T0(M; ffl) is an isomorp*
*hism (cf. [S,
Prop. 3.9.7]). Here 0 is the trivial elementary abelian p - group and ffl is th*
*e augmentation
of the connected algebra K. In particular, we see that in this case the modules*
* JK (n) and
I(0;ffl)(n) agree.
If K is noetherian, then it is easy to check that the modules I(V;')(n) are fin*
*itely generated
K - modules for any (V; ') 2 R(K). The following result shows that these modul*
*es give
enough injectives in the category Kfg - U. We will give two proofs of this resu*
*lt which both
rely crucially on the main result of [H1].
THEOREM 1.9. (Existence of enough injectives). Let K be an unstable noetherian *
*algebra
and M an unstable finitely generated K - module. Then there is an embedding M -*
*! I in
the category Kfg - U in which I is isomorphic to a finite direct product of inj*
*ective modules
I(V;')(n) for suitable (V; ') in R(K) and natural numbers n.
First_Proof_of_1.9:____From Theorem I of [H1] we know that there is a finite fi*
*ltration 0 = M0
M1 ::: Mn = M such that the successive filtration quotients Mi=Mi-1 are ki - *
*fold
suspensions of modules Li which can be embedded (in the category Kfg - U) into *
*a finite
direct product of modules H*BV (') = I(V;')(0). It is enough to show that the *
*theorem
holds for the quotients Mi=Mi-1. Now Mi=Mi-1 can be embedded into a finite dire*
*ct sum
of modules H*BV (') kiFp. The K - module structure on H*BV (') kiFp is clearly
pulled back from the obvious H*BV TV (K; ') - module structure via the map flK*
*;(V;'):
K -! H*BV TV (K; '). Now H*BV kiFp can be embedded (as H*BV TV (K; ')
- module) into H*BV JTV(K;')(ki), hence Mi=Mi-1 can be embedded_in Kfg - U int*
*o a
finite direct product of modules I(V;')(ki) and we are done. |_|
The second proof relies on the following result of [HLS2] which again depends o*
*n [H1,Thm.I].
THEOREM 1.10. Let K be an unstable noetherian algebra and M an unstable finitel*
*y gen-
erated K - module. Then there is a natural number n such that the maps flM;(V;'*
*)(which are
adjoint to the projection maps TV M -! TV (M; ')) induce an embedding
Y
fl : M -! H*BV TV (M; ') 0} i*
*s an
unstable K - submodule which we denote by FaM. In fact, this assignment extend*
*s to a
functor Fa from Kfg - U to itself which we call the a - torsion submodule funct*
*or. It is left
exact and its right derived functors are denoted by RiFa. If a = K+ we will wri*
*te F instead
of FK+ and call F the "finite submodule functor".
Proof:______The only part which is not obvious is that FaM is closed under Stee*
*nrod operations.
For simplicity assume that the prime is 2. Let x 2 FaM, a 2 a, so anx = 0 for s*
*ome large
n. We may assume that n = 2k and kkis large. Then we apply the total Steenrod o*
*peration
to this equation and obtain Sq(a)2kSq(x) = 0. Now assume that i < 2k and consi*
*derkthe
homogeneous part of degree |a2 x| + i (|y| denoting the degree of y). We obtain*
*_a2 Sqix = 0,
i.e. Sqix is again a - torsion. The argument for odd primes is analogous. |_|
The final result in this section together with the embedding results of 1.9 and*
* 1.10 shows
that primary decompositions (cf. [L]) exist in Kfg - U if K is noetherian. We*
* leave it to
the interested reader to state and prove the appropriate existence and uniquene*
*ss results for
such decompositions.
PROPOSITION 2.15. Assume K is an unstable algebra and M is an unstable K - modu*
*le.
Assume we have a map M -! H*BV (') F of unstable K - modules with F a finite
TV (K; ') - module for some (V; ') 2 R(K). Then the kernel of this map is prim*
*ary with
respect to the prime ideal Rad (Ker '). Furthermore, if F vanishes in degrees n*
* and bigger,
then (Ker ')n kills H*BV (') F .
Proof:______To see this take an element x 2 K and consider flK;(V;')(x) 2 H*BV *
*TV (K; '). This
can be written as flK;(V;')(x) = 'x 1 + y with y 2 H*BV TV (K; ')+and TV (K; *
*')+
denoting as before the ideal of elements of positive degree. From this formula *
*it is clear that
x acts nilpotently on H*BV (') F iff x 2 Rad_(Ker '), and if x1; : :;:xn are i*
*n Ker ', then
the product x1 : :x:nkills H*BV (') F . |_|
3. Localizations in Kfg - U away from torsion modules
3.1.______Throughout this section K will be an unstable noetherian algebra. We *
*will study local-
izations on the category Kfg - U of unstable finitely generated K - modules awa*
*y from the
14
full subcategory T ors(a) of a - torsion modules for some fixed invariant ideal*
* a. We begin
with some formal definitions.
DEFINITION 3.2. Let K be an unstable noetherian algebra and let a be an invaria*
*nt ideal of
K. Let M be an unstable finitely generated K - module.
a) M is called T ors(a) - reduced (or a - reduced) iff Hom K-U (N; M) = 0 for e*
*ach a - torsion
module N 2 Kfg - U.
b) M is called T ors(a) - closed (or a - closed) iff ExtiK-U(N; M) = 0, i = 0; *
*1 for all a -
torsion modules N 2 Kfg - U.
It is clear that M is a - reduced iff it does not contain any non-trivial a - t*
*orsion submodules.
Furthermore, M is a - closed iff for any morphism ff : A -! B of unstable finit*
*ely generated
K - modules whose kernel and cokernel is a - torsion, the induced map Hom K-U (*
*B; M) -!
Hom K-U (A; M) is an isomorphism.
The following proposition follows immediately from the definitions (cf. [G], or*
* [HLS1,2] where
the same concept has been investigated in other settings).
PROPOSITION 3.3. Let K be an unstable noetherian algebra and let a be an invari*
*ant ideal.
a) If 0 -! M -! M1 -! M2 is exact and M1 is a - closed and M2 is a - reduced th*
*en M
is a - closed.
b) Any finite inverse limit of a - closed modules is a - closed.
_
c) Any summand of an a - closed module is a - closed. |_|
3.4.______The categories T ors(a) are "Serre subcategories", i.e. their set of *
*objects form a Serre
class. They are localizing in the sense of Gabriel [G, III.3. Cor. 1] so the*
*re are functors
La : Kfg - U ! Kfg - U and natural transformations a : 1Kfg-U ! La such that fo*
*r each
M 2 Kfg - U:
o LaM is a - closed and
o kernel and cokernel of a;M are a - torsion.
We will study the localization away from these subcategories. The following res*
*ult provides
general examples for a - closed modules.
PROPOSITION 3.5. Let K be an unstable noetherian algebra, let a be an invariant*
* ideal and
(V; ') 2 R(K) be such that Rad (Ker ') does not contain a, in other words (V; '*
*) 2 O(a).
Then the following assertions hold for each unstable finitely generated K - mod*
*ule M.
a) ExtiK-U(N; TV (M; ')) = 0 for each a - torsion module N in Kfg - U and each *
*i. In
particular TV (M; ') is a - closed.
b) RiFa(TV (M; ')) = 0 for all i.
The proof of the proposition relies on the following key lemma.
15
LEMMA 3.6. Let F be an unstable finite TW (K; ) - module for some (W; ) 2 R(K*
*) and
let (V; ') be in R(K). Then there is an isomorphism of unstable K - modules
Y
TV (H*BW ( ) F ; ') ~= H*BW ( ) F :
HomR(K)((V;');(W; ))
Proof_of_Lemma_3.6:______By definition TV (H*BW ( )F ; ') ~=Fp(')T0VKTV (H*BW (*
* )F ).
Futhermore, because F is finite and TV commutes with tensor products we have an*
* isomor-
phism of unstable modules (see [L2,L3])
Y
TV (H*BW F ; ') ~= H*BW F ;
Hom(V;W)
and we have to identify the TV K - module structure, and in particular the TV0K*
* - mod-
ule structure on this. Now F is bounded above, so the action of TW (K; ) on F*
* factors
through an action of (TW (K; )) 0. In particular, limiO(a)TV (M; ') is an unstable finitely genera*
*ted a - torsion
module for all i > 0.
The following special case of this theorem needs to be emphasized.
COROLLARY 3.10. Let K be an unstable noetherian algebra and let M be an unstabl*
*e finitely
generated K - module.
a) Then there is a natural exact sequence
0 -! F M -! M -ae!limR*(K)TV (M; ') -! R1F M -! 0
17
in which the components of ae are induced by the maps aeM;(V;'). In particular,*
* the kernel and
cokernel of ae are finite. Furthermore, ae is localization away from the subca*
*tegory of finite
unstable K - modules.
b) There are natural isomorphisms
limiR*(K)TV (M; ') ~=Ri+1F M
_
for all i > 0. In particular, limiR*(K)TV (M; ') is finite for all i > 0. |_|
3.11._Remarks.______a) Theorem 0.4 and Corollary 0.5 of the introduction are cl*
*early just special
cases of 3.9 and 3.10: the two subsets resp. subcategories both labelled O(a)*
* (of A(G)
and R(H*BG) respectively) clearly correspond under the equivalence of categorie*
*s A(G) ~=
R(H*BG) of 0.2.I; furthermore for M = H*GX the isomorphisms of 0.2.II are compa*
*tible with
the maps aeM;(V;')and the maps H*GX -! H*CG(E)(XE ) induced by the inclusions. *
*This is
obvious from the construction of the isomorphisms (as desribed in the appendix).
b) Of course, as in the case of Theorem 0.4 of the introduction we have here th*
*e same formal
analogy with the classical case considered in [Gr].
c) If a is the ideal of positive dimensional elements of K then there is a vani*
*shing result for the
higher limits due to Oliver [O]: limiR*(K)TV (M; ') = 0 if i > d(K) where d(K) *
*is the maximal
rank of an elementary abelian p - group V such that (V; ') 2 R(K) for some ' : *
*K -! H*V .
This is analogous to the vanishing theorem of Grothendieck [Ha, Theorem III.2.7*
*]. However,
for more general ideals this analogy breaks down: e.g. if O is the open (!) set*
* consisting of
all (V; ') with Rad(Ker ') equal to a fixed minimal prime ideal then O = O(a) w*
*here a is the
intersection of all other invariant primes. In this case O(a) is equivalent to *
*the one-object-
category associated to the automorphism group Aut R(K)((V; ')) and the higher l*
*imits can
be identified with group cohomology and can be non-zero in arbitrary high degre*
*es.
The following result represents the key step in the proof of Theorem 3.9.
LEMMA 3.12. Let K, a and O(a) be as in 3.9, let (W; ) be in R(K), F be an unst*
*able finite
TW (K; ) - module and consider the unstable K - module M = H*BW ( ) F .
a) If (W; ) is in O(a) then
ae : M -! limO(a)TV (M; ')
is an isomorphism and for each i > 0
limiO(a)TV (M; ') = 0 :
b) If (W; ) is not in O(a) then for each i 0
limiO(a)TV (M; ') = 0 :
18
Proof:______By Lemma 3.6Qthe functor on O(a) which sends (V; ') to TV (M; ') is*
* given by
TV (H*BW ( ) F ; ') ~= HomR(K)((V;');(W;())H*BW ( ) F ). In other words it*
* is in-
duced from the graded vector space H*BW ( ) F if (W; ) 2 O, and trivial other*
*wise.
(Here we identify the category of graded vector spaces with the category of fun*
*ctors from the
"trivial category", with one object (W; ) and the_identity morphism only, to t*
*he category of
graded vector spaces.) The Proposition follows. |_|
Proof_of_Theorem_3.9:_____Consider an injective resolution Io of M in Kfg - U a*
*s provided by
Theorem 1.9 and Corollary 1.11. Because T is exact the complex of functors T- (*
*Io; -) is a
resolution of the functor T- (M; -). By the previous proposition the higher der*
*ived functors
of limO(a) vanish on the functors T- (Ik; -). This together with the fact that*
* limO(a) is
left exact implies that limiO(a)T- (M; -) can be computed as the cohomolgy of t*
*he complex
lim OT- (Io; -)
Again by the previous proposition the complex limOT- (Io; -) is obtained from I*
*o by throwing
away those I(V;')(n) for which (V; ') is not in O(a) and keeping all the others*
*. In other words,
we get an exact sequence of complexes
0 -! FaIo -! Io -! limO(a)TV (Io; ') -! 0 :
The long exact sequence belonging to this short exact sequence yields the exact*
* sequence of
a) and the isomorphisms in b).
_
Finally the inverse limit is a - closed by 3.3 and 3.5. |_|
3.13.______The proof of 3.9 together with 2.15 gives also information about the*
* height of the a -
torsion modules RiFaM in terms of a given injective resolution. For example, if*
* p = 2 and if
ni is the maximum of all n such that for some (V; ') 2 C(a) the injective I(V;'*
*)(n) occurs as
a summand in Ik , then the height of the a - torsion module RiFaM is at most nk.
In the case of equivariant cohomology information about injective resolutions i*
*s often available
(see [HLS2, II.1 and II.2], [HLO]). In joint work with F.Cohen and Y. Xia we wi*
*ll apply this
in the case of mapping class groups to get new classes in their cohomology.
Theorem 3.9 together with Theorem 0.2 and the obvious generalization of Corolla*
*ry 3.8 imply
the following result. It contains the theorem of Jackowski and McClure [JM] (se*
*e also [DW2])
as a special case. (Take O = A*(G) and C be isomorphic to Z=p!) Note also that *
*under the
equivalence of categories of Theorem 0.2.I a subset O of A(G) will be called op*
*en if E 2 O
implies E0 2 O whenever E is subconjugate to E0.
COROLLARY 3.14. Let p be a prime. Assume G is a compact Lie group and let C b*
*e an
elementary abelian p - subgroup of G which is central in some p - Sylow subgrou*
*p. Let O be
any open subset of A(G) containing C.
19
a) Then the restriction maps H*BG -! H*BCG (E) induce an isomorphism
ae : H*BG -! limOH*BCG (E) :
_
b) Furthermore limiOH*BCG (E) = 0 for all i > 0. |_|
The following change of rings type result gives us some flexibility for computi*
*ng the higher
limits. In particular it allows a change of groups in the situation of equivari*
*ant cohomology.
The result is implicit in [DW2].
PROPOSITION 3.15. Let K, L be unstable noetherian algebras and let f : K -! L *
*be a
homomorphism of unstable algebras which makes L into an unstable finitely gener*
*ated K -
module. Let M be an object in Lfg - U which we also consider as an object of Kf*
*g - U via
the map f. Then for any open set O 2 R(K) there are natural isomorphisms for al*
*l i > 0
limiOTV (M; ') ~=limif*-1OTV (M; ) :
(Of course, f* denotes the induced map R(L) -! R(K) and f*-1 O is the full subc*
*ategory
of R(L) whose objects are mapped to O under f*.)
Proof:______The map f induces a functor f* : f*-1 O -! O and a functor f! : O -*
* mod -!
f*-1 O - mod . Here we write D - mod for the category of functors from a cate*
*gory D
to the category of abelian groups. The objects of D - mod are called D - modu*
*les. Let
fe!: f*-1 O - mod -! O - mod be the right Kan - extension along f!so that we ha*
*ve natural
isomorphisms
Hom O-mod (F; ef!G) ~=Hom f*-1O-mod (f!F; G)
for any O - module F and any f*-1 O - module G.
We recall that the right Kan - extension of an f*-1 O - module G is given on th*
*e object
(V; ') 2 O as follows: Let (V; ') # f! be the under category with respect to f*
*!. Then
(ef!G)(V; ') = lim(V;')#f!G.
We will need the following lemma. We omit its proof which is not difficult and *
*completely
analogous to that of Lemma 2.9 of [H3].
LEMMA 3.16. Let (V; ') be an object in O. Then the under category (V; ') # f! i*
*s a disjoint
union of categories each of which has an initial object. The components are ind*
*exed by those
(V; ) 2 f*-1 O which extend (V; '), i.e. for which_' = f holds. These objects*
* are at the
same time the initial objects of the components. |_|
The lemma implies that the Kan extension is given by the following formula:
Y
(ef!G)(V; ') = G(V; ) :
(V; )
20
The product in this formula is indexed by those (V; ) 2 f*-1 O which extend (V*
*; '). In
particular, the Kan - extension ef!is exact. The functor f! is clearly exact an*
*d we conclude
that it carries projective resolutions to projective resolutions. Taking a proj*
*ective resolution
of the constant functor with value Z we obtain for any R(L) - module G:
limiOef!G ~=limif*-1OG :
Now let M be an unstable L - module and consider the f*-1 O - module T- (M; -).*
* Now
let (V; ') 2 O. It is easy to check that (with an obvious abuse of notation) w*
*e have
(f!T- (M; -))(V; ') ~=TV_(M; '). In other words, we obtain the right Kan - ext*
*ension and
the result follows. |_|
4. An application to H*BGL(n; Z[1=2])
4.1.______We give an application of 3.9 to a qualitative study of the mod - 2 c*
*ohomology ring
H*BGL(n; Z[1=2]). Here GL(n; Z[1=2]) is the general linear group of rank n ove*
*r the ring
Z[1=2]. Let Dn be the subgroup of diagonal matrices with diagonal entries 1. Th*
*e image
of the restriction map aen : H*BGL(n; Z[1=2]) -! H*BDn has recently been determ*
*ined by
Mitchell [Mt]; it is isomorphic to a tensor product F2[w1; : :;:wn]E(e1; : :;:e*
*2n-1) of a poly-
nomial algebra on generators w1; : :;:wn and an exterior algebra on generators *
*e1; : :;:e2n-1,
with indices giving the degrees of the elements. Let n0 be minimal such that ae*
*n0 is not in-
jective. According to Dwyer (private communication) n0 is finite. We will show *
*in [H4] that
n0 > 3. The theory developed here can be used to get some qualitative informati*
*on about
the size of Ker aen.
To do this we recall (cf. [HLS2,II.5.2]) that the objects of Rn := R(H*BGL(n; *
*Z[1=2]))
can be identified with the faithful representations of elementary abelian 2 - g*
*roups V , i.e.
with formal sums OnOO where O runs through the characters of V and the nO are *
*non-
negative integers such that OnO = n and such that the set of O with nO > 0 span*
*s V *,
the group of characters of V . (The representation ' is identified with the obj*
*ect (V; '*) in
R(H*BGL(n; Z[1=2])), if '* denotes the induced map in cohomology, and correspon*
*ds to the
object Im ae in A(GL(n; Z[1=2])) under the equivalence of categories in 0.2.I.)*
* Furthermore,
if ' = OnOO (resp. '0 = O0nO0O0) are faithful representations of V (resp. V *
*0) then
Hom Rn ('; '0)Q= {ff 2 Hom (V; V 0)|' = O0nO0O0ff}. We note that the centraliz*
*er of Im ' is
isomorphic to OGL(nO; Z[1=2]).
Let On Rn be the subset consisting of those representations ' for which nO < n*
*0 for all O.
This is clearly an open subset and we have On = O(an) where an is the invariant*
* ideal given
asQthe intersection of all Rad (Ker '*) with (V; '*) =2On. ThenQTV (H*BGL(n; Z[*
*1=2]); ') ~=
O H*BGL(nO; Z[1=2]) by Theorem 0.2 and hence embedds into OH*BDnO ~= H*BDn
for all ' 2 On. Consequently the kernel of the localization map away from T ors*
*(an) modules
agrees with the kernel of aen. In particular, Ker aen is an - torsion. If ' i*
*s a representation
21
of V which is not in On then V has rank at most equal to n - n0 + 1, therefore *
*the size (as
defined in the introduction) of H*BGL(n; Z[1=2])= Rad(an) is at most equal to n*
* - n0 + 1.
Clearly this number is an upper bound for the size of the an - torsion module K*
*er aen0. In
fact, we have the following theorem.
THEOREM 4.2. The kernel of the restriction map H*BGL(n; Z[1=2]) -aen!H*BDn is *
*the
largest an - torsion submodule and for n n0 it has size n - n0 + 1 where n0 de*
*notes the
smallest natural number such that aen0 is not a monomorphism. In particular, t*
*he size of
Ker aen0 is one, i.e. Ker aen0 is periodic in large degrees.
Proof:______We have already seen that the size is at most n - n0 + 1.
Let dn be equal to the size of Ker aen. Consider an embedding of Ker aen0 as i*
*n 1.9, and
assume it is minimal in the sense that no factor I(V;')(n) can be dropped witho*
*ut loosing
the embedding property. Then it is clear that the size of Ker aen is equal to *
*the maximum
of the sizes of the injectives I(V;')(n) involved, and hence exactness of T and*
* Lemma 3.6.
imply that each component TV (Ker aen; '*) has size at most dn. It suffices the*
*refore to find a
faithful representation ' : V - ! GL(n; Z[1=2]) such that TV (Ker aen; '*) has *
*size n - n0 + 1.
Such a representation can be obtained as follows. Let V be elementary abelian*
* of rank
n-n0+1 and let Oi, 1 i n-n0+1 be a dual basis of V . Then consider ' = n0O1+i*
*6=1Oi.
Applying TV (-; '*) to the exact sequence
0 -! Keraen -! H*BGL(n; Z[1=2]) -! H*BDn
yields an exact sequence
Y
0 -! TV (Ker aen; '*) -! H*BGL(n0; Z[1=2]) H*BDn-n0 -! H*BDn
where the product is taken over all homomorphisms : V - ! Dn such that its co*
*mposition
with the inclusion of Dn into GL(n; Z[1=2]) is conjugate to ' (cp. the discussi*
*on of the Kan
extension in the proof of 3.15.) Now the different maps H*BGL(n0; Z[1=2])H*BDn-*
*n0 -!
H*BDn differ only by the action of an appropriate element of the symmetric grou*
*p Sn on
the target, in particular all these maps have the same kernel, which is equal t*
*o Ker aen0
H*BDn-n0. This has size n - n0 + dn0 and hence we only have to show that the s*
*ize of
Ker aen0 is positive. However, Theorem 0.2 and Proposition 3.5 show that H*BGL(*
*n; Z[1=2])
does not contain any unstable finite ideals (take for a the ideal of elements i*
*n positive_degrees
and for ' the restriction map to the central Z=2), so the size must be positive*
*. |_|
The method used in the discussion above should lead to similar results for gene*
*ral linear
groups over rings of S - integers in other number fields.
22
Appendix. Lannes' T - functor and Borel constructions of discrete groups
Let p be a fixed prime. As before we surpress the coefficients from our notatio*
*n.
In this appendix we prove part (b) of Theorem 0.4. We will use freely results a*
*nd terminology
of [HLS1].
Let V be an elementary abelian p - group and ae a homomorphism from V to the (d*
*iscrete)
group . Let X be a - space. Denote the centralizer of the image of ae in by a*
*eand the
fixed point set with respect to the image of ae by Xae. Then the homomorphism V*
* x ae! ,
(v; g) 7! ae(v)g induces a map BV x (Eaexae Xae) -! E x X which we denote by c*
*ae.
Passing to cohomology and using adjointness we obtain a map ad(c*ae) : TV H*X -*
*! H*aeXae.
THEOREM A.1. Let be a discrete group and X a finite dimensional - CW - comple*
*x of
finite orbit type whose isotropy groups are all finite. Then the natural map
Y
TV (H*X) -! H*ae(Xae)
ae2Rep(V;)
with components ad(c*ae) is an isomorphism for each elementary abelian p - grou*
*p V . (Here
Rep (V; ) denotes the set of - conjugacy classes of homomorphisms from V to *
*and we
have chosen a representative ae from each conjugacy class. )
Proof:______Because TV is exact and commutes with direct sums, it is enough to *
*do the case of an
orbit, i.e. X = G=H with H finite. In this case we have natural isomorphisms
Y
TV H*X ~=TV H*BH ~= H*BHae
ae2Rep(V;H)
where the second isomorphism comes from Lannes' Theorem [L1,L3] and has compone*
*nts
ad(c*ae) as in the statement of the theorem. Now it follows from [H3, Lemma 2.8*
*] that there
is a natural isomorphism
Y Y
(A:1:1:) H*BCH (ae) ~= H*ae(Xae) :
ae2Rep(V;H) ae2Rep(V;)
It is straightforward_to check that this string of isomorphisms is given by the*
* natural map in
question. |_|
If X can also be chosen mod - p acyclic then by Smith theory the fixed point se*
*ts Xaeare
mod - p acyclic as well. Hence we have H*X ~=H*B and H*aeXae~=H*Bae. Consequent*
*ly
we obtain the following result.
23
COROLLARY A.2. Let be a discrete group which admits an action on a finite dime*
*nsional
mod - p acyclic - CW - complex with finite orbit type and with finite isotropy*
* groups. Then
the natural map Y
TV (H*B ) -! H*(Bae)
ae2Rep(V;)
_
is an isomorphism for each elementary abelian p - group V . |_|
THEOREM A.3. Let be a discrete group and X be a - CW - complex with finitely *
*many
equivariant cells and finite isotropy groups. Then H*(X) is a finitely generate*
*d algebra.
Proof:______We will first prove that H*(X) is noetherian up to F - isomorphism,*
* or more precisely,
that its Nil - closure is noetherian.
Consider the contravariant functor g(H*X) from elementary abelian p - groups to*
* sets which
associates to an elementary abelian p - group V the set Hom K (H*X; H*BV ). T*
*his set is
given by the spectrum of the p - Boolean algebra TV0H*X [L2,L3] which by Theore*
*m A.1
above can be identified with the disjoint union
a a
ss0(Eaexae Xae) ~= ss0(Xae)=ae:
ae2Rep(V;) ae2Rep(V;)
It follows from (A.1.1) together with our finiteness assumptions that this set *
*is finite (although
the set Rep (V; ) need not be finite). The functor g(H*X) has finite transcenda*
*nce degree
d in the sense of [HLS1,II.5] and, in fact, d is equal to the maximal rank of a*
*n elementary
abelian subgroup which occurs as isotropy subgroup in X. Furthermore the End (*
*V ) - set
g(H*X)(V ) is noetherian in the sense of [HLS1], which means that the Nil - clo*
*sure of H*X
(which is given, up to isomorphism, by Quillen's inverse limit) is noetherian [*
*HLS1,II.7].
We can now pick an unstable noetherian subalgebra K of H*X which is F - isomorp*
*hic to
the Nil - closure of H*X. The spectral sequence of the map E x X -! \X is a sp*
*ectral
sequence of H*X - modules and hence one of K - modules. Because K is noetherian*
* and this
spectral sequence has only finitely many columns it is enough toLshow that its *
*E1 - term is a
finitely generated K - module. The E1 - term is given as Es;*1= oeH*(Boe) whe*
*re oe runs
over the set of s - cells in \X and oedenotes the isotropy group of a chosen re*
*presentative
of the set of cells in X which project to the cell oe in \X. It is clearly enou*
*gh to consider
a single cell, i.e. to show that H*(Boe) is a finitely_generated K - module. Ho*
*wever, this
follows immediately from [HLS1, II. Prop. 7.8]. |_|
In particular, if X can also be chosen mod - p acyclic we obtain.
COROLLARY A.4. Let be a discrete group which acts on a mod - p acyclic - CW -
complex with finitely_many - cells and with finite isotropy groups. Then H*B i*
*s a finitely
generated algebra. |_|
24
We repeat that interesting examples of such groups are (S)-arithmetic groups, m*
*apping class
groups, outer automorphism groups of free groups and word-hyperbolic groups in *
*the sense
of Gromov.
Proof_of_Theorem_0.2.b_____: The functor g(H*B) which by A.2 sends V to Rep(V; *
*) determines
the category R(H*B) (see [HLS1,II.7]); as in the other cases of Theorem 0.2 its*
* objects can
be identified with the conjugacy classes of monomorphisms from elementary abeli*
*an p - groups
to . This category is (for any group ) equivalent to the category A() and part *
*I follows.
_
Part II follows immediately from Theorem A.1. |_|
A.5_Remark._______In [H3] we use that Corollary 0.5 of this paper holds under t*
*he assumptions
of Theorem A.3 which are a bit more general than those of 0.5 because it is not*
* assumed
that H*B is noetherian. However, by an argument as in 3.15 one shows that Theo*
*rem
A.1 implies limiA*()H*C (E)(XE ) ~=limiR*(H*X)TV (H*X; ') and then 3.10 gives t*
*he more
general case as well.
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26