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 compact Lie - group 
(not necessarily connected) or a discrete group of finite virtual 
cohomological dimension (f.v.c.d.  for short), e.g.  the
general linear group over the ring of S -integers in a number 
field or the mapping class group of an orientable surface. In [Q1] 
Quillen considered the category A(G) whose objects are the
elementary abelian p - subgroups of G and whose morphisms are given 
as compositions 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 
nilpotent elements and such that for each element in the limit a 
sufficiently large pn - th power is 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 Steenrod 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 Quillen's map
qG and ending with a genuine isomorphism. To do this we consider the 
full subcategory Niln of the category U of all unstable modules; 
Niln is the smallest subcategory of U which contains all n - fold 
suspensions and isclosed with respect to forming extensions and taking
filtered colimits. Here is an explanation for our terminology. The 
unstable module underlying an unstable algebra (e.g. the 
mod p - cohomology of a space) is an object in Nil1 if andonly if
all its elements are nilpotent in the usual sense. The subcategory 
Niln is localizing, i.e. there exist a localization functor Ln and 
a natural transformation n : idU ! Ln ,the localization
away from Niln. Quillen's map qG is actually localization away from Nil1. 
The sequence of approximations referred to above will be the sequence 
of localizations away from Niln and the following result shows that 
this 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 large n.