Localization and duality in topology and modular representation theory
David J. Benson and J. P. C. Greenlees

\begin{abstract}
We develop a duality theory for localizations in
the context of ring spectra in algebraic topology. We apply
this to prove a theorem in the modular representation theory
of finite groups.

Let $G$ be a finite group and $k$ be an algebraically closed field of
characteristic $p$. If $\p$ is a homogeneous nonmaximal prime ideal in
$H^*(G,k)$, then there is an idempotent module $\kappa_\p$ which
picks out the layer of the stable module category corresponding to
$\p$, and which was used by Benson, Carlson and Rickard
\cite{Benson/Carlson/Rickard:1997a} in their development of
varieties for infinitely generated $kG$-modules. Our main theorem
states that the Tate cohomology
$\hat H^*(G,\kappa_\p)$ is a shift of the injective hull of
$H^*(G,k)/\p$ as a graded $H^*(G,k)$-module. Since $\kappa_\p$
can be constructed using a version of the
stable Koszul complex, this can be viewed
as a statement of localized Gorenstein duality in modular representation
theory. Various consequences of this theorem are given, including
the statement that the stable endomorphism ring of
the module $\kappa_\p$ is the $\p$-completion of cohomology
$H^*(G,k)\phat$, and the statement that $\kappa_\p$ is
a pure injective $kG$-module.

In the course of proving the theorem, we further develop the framework
introduced by \DGI\ for translating between the unbounded derived categories
$\D(kG)$ and $\D(C^*(BG;k))$. We also construct a functor $\Psi\colon
\D(kG) \ra \StMod(kG)$ to the full stable module category, which
extends the usual functor $\D^b(kG) \ra \stmod(kG)$ and which preserves
Tate cohomology. The main theorem is formulated and proved in $\D(C^*(BG;k))$,
and then translated to $\D(kG)$ and finally to $\StMod(kG)$.

The main theorem
in $\D(C^*(BG;k))$ can be viewed as stating that a version of
Gorenstein duality holds after localizing at a prime ideal in $H^*(BG;k)$.
This version of the theorem holds more generally for a compact Lie group
satisfying a mild orientation condition. This duality lies behind the
local cohomology spectral sequence of Greenlees and Lyubeznik
for localizations of $H^*(BG;k)$.\vspace{-5mm}
\end{abstract}