\title{Complexes of injective $kG$-modules} 

\author{Dave Benson}
\address{Department of Mathematics, University of Aberdeen, Aberdeen
AB24 3EA, Scotland, UK}

\author{Henning Krause} 
\address{Institut f\"ur Mathematik, Universit\"at Paderborn,
33095 Paderborn, Germany}


\begin{abstract}
Let $G$ be a finite group and $k$ be a field of characteristic $p$.
We investigate the homotopy category $\KInj{kG}$ of the
category $\CInj{kG}$ of complexes of
injective ($=$ projective) $kG$-modules. If $G$ is a $p$-group, this
category is equivalent to the derived category $\Ddg(C^*(BG;k))$
of the cochains on the classifying space; if $G$ is not a $p$-group
it has better properties than this derived category. The 
ordinary tensor product in $\KInj{kG}$ with diagonal $G$-action
corresponds to the $E_\infty$ tensor product on $\Ddg(C^*(BG;k))$.

We show that $\KInj{kG}$ can be regarded as a slight enlargement of
the stable module category $\StMod(kG)$. It better formal
properties inasmuch as the ordinary cohomology ring $H^*(G,k)$ is
better behaved than the Tate cohomology ring $\hat H^*(G,k)$. 

It is also better than the derived category $\D(\Mod kG)$, because
the compact objects in $\KInj{kG}$ form a copy of the
bounded derived category $\D^b(\mmod kG)$, whereas the compact
objects in $\D(\Mod kG)$ do not include objects corresponding
to the finitely generated $kG$-modules.

Finally, we develop the theory of support varieties and homotopy colimits
in $\KInj{kG}$.
\end{abstract}