\magnification=\magstep1
%\input foot.tex
\input macro.tex
\input pop-foot.tex
%\magnification=\magstep1
\baselineskip13pt
\hsize=17truecm
\vsize=23truecm
\hoffset=-0.5truecm
\parindent=0cm
\overfullrule=0mm
%\rightline{\number\day.\number\month.\number\year}
\bigskip
\centerline{\bf COMMUTATIVE ALGEBRA OF UNSTABLE $K$ - MODULES,}
\bigskip
\centerline{\bf LANNES' $T$ - FUNCTOR AND}
\bigskip
\centerline{\bf EQUIVARIANT MOD - P COHOMOLOGY}
\bigskip
%\centerline{(Preliminary version)}
%\bigskip
\centerline{by}\bigskip\bigskip
\centerline{Hans--Werner Henn}
\bigskip\bigskip
\bigskip
{\baselineskip=10pt\maimic
\centerline{\bf Abstract}
\bigskip
\vbox{\parshape=1 0.5truecm 15.3truecm
%\baselineskip=10pt\maimic
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].}
\smallskip
\vbox{\parshape=1 0.5truecm 15.3truecm
%\baselineskip=10pt\maimic
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.}}
\parshape=0
\smallskip
\bigskip\bigskip
\end