COMMUTATIVE ALGEBRA OF UNSTABLE $K$ - MODULES, LANNES' $T$ - FUNCTOR AND
EQUIVARIANT MOD - P COHOMOLOGY
Hans--Werner Henn

Abstract

Let $p$ be a fixed prime and  
let $K$ be an unstable algebra over the mod - $p$ Steenrod algebra $A$ 
such that $K$ is finitely generated as graded $\FF_p$ - algebra. 
Let $K_{fg}-\Ua$ denote the abelian category of finitely generated 
$K$ - modules with a compatible unstable $A$ - module structure.
We study various concepts of commutative algebra in this setting. 
The r\^ole of the prime ideal spectrum of a commutative ring is here 
taken by a category $\Rav (K)$ which, roughly speaking, consists of 
the $A$ - invariant prime ideals of $K$ together with certain 
``Galois information''; sheafs will correspond to functors on this 
category, and the r\^ole of the sheaf associated to a module will 
be taken by the components of Lannes' $T$ - functor. We discuss the 
notions of support, of ${ \gl a}$ - torsion modules 
(for an invariant ideal ${ \gl a}$ of $K$) and of localization away 
from the Serre subcategory $\Ta ors ({ \gl a})$ of ${ \gl a}$ - torsion 
modules in our setting. We show that the category $K_{fg}-\Ua$ has 
enough injectives and use these injectives to study these localizations 
and their derived functors; they are closely related to the derived 
functors of the ${ \gl a}$ - torsion functor $F_{{ \gl a}}$. 
Our results are formally analogous to Grothendieck's results in the 
classical situation of modules over a noetherian commutative ring R [Gr].} 
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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 $\Rav (K)$ 
and the functor on $\Rav (K)$ associated 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 the 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 $D_n$ of diagonal matrices.}}