16.3.1993
LOCALIZATIONS OF UNSTABLE A - MODULES
AND EQUIVARIANT MOD p COHOMOLOGY
by
Hans-Werner Henn, Jean Lannes and Lionel Schwartz
Introduction
Let p be a fixed prime and G be either a compactLie - group (not necessarily co*
*nnected)
or a discrete group of finite virtual cohomological dimension (f.v.c.d. for sh*
*ort), e.g. the
general linear group over the ring of S -integers in a number field or the mapp*
*ing class group
of an orientable surface.In [Q1] Quillen considered the category A(G) whose obj*
*ects are the
elementary abelian p - subgroups of G and whose morphisms are given as composit*
*ions of
inclusions and conjugations. He showed that the restriction homomorphisms from *
*the mod
p cohomology H BG of the classifying space BG to H BE for E 2 A(G)induce a map
qG : H BG ! limA(G)opH BE such that the kernel of qG consists of nilpotentelem*
*ents
and such that for each element in the limit a sufficiently large pn - th poweri*
*s in the image
of qG. (As usual, A(G)op denotes theopp osite category of A(G), so that E 7! H *
*BE is a
covariant functor on A(G)op.)
In this paper we will consider H BG as an unstable module over the mod p Steenr*
*od algebra
A (unstable module for short) and show how this structure can be used to refine*
* Quillen's
result and describe a finite sequence of approximations to H BG starting with Q*
*uillen's map
qG and ending with a genuine isomorphism. To do this we consider the full subca*
*tegory
Niln of the category U of all unstable modules; Niln is the smallest subcategor*
*yof U which
contains all n - fold suspensions and isclosed with respect to forming extensio*
*ns and taking
filtered colimits. Here is an explanation for our terminology. The unstable mod*
*ule underlying
an unstable algebra (e.g. the mod p - cohomology of a space) is an object in Ni*
*l1 if andonly if
all its elements are nilpotent in the usual sense. The subcategory Niln is loca*
*lizing,i.e. there
exist a localization functor Lnand a natural transformation n : idU ! Ln ,the l*
*ocalization
away from Niln. Quillen's map qG is actually localization away from Nil1. The s*
*equence of
approximations referred to abovewill be the sequence of localizations away from*
* Niln and
the following result shows thatthis sequence of localizations is in many cases *
*actually finite.
THEOREM 0.1[He]. Let K be an unstable algebra which is finitely generated as an*
* algebra.
Then the localization away from Niln is an isomorphism for all sufficiently lar*
*ge n.
The localization LnH BG and the localization map ncan be roughly described as f*
*ollows.
As noted above the map 1arose from the product of restriction homomorphismsH B*
*G !
Q Q
E2A(G)H BE, and L1H BG was given bythe subalgebra of E2A(G)H BE consisting
of those families of elements fxEgE2A(G) which satisfy the compatibility condit*
*ions imposed
by the morphisms in Quillen's category A(G). The description of n is similar. L*
*et CG(E)
denote the centralizer of Ein G and (H BCG (E))