Complexes of injective kG-modules

Abstract

Let $G$ be a finite group and $k$ be a field of characteristic $p$. We investigate the homotopy category $K\left(InjkG\right)$ of the category $C\left(InjkG\right)$ of complexes of injective ($=$ projective) $kG$-modules. If $G$ is a $p$-group, this category is equivalent to the derived category $D_{dg}\left(C^{\ast }\left(BG;k\right)\right)$ 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 $K\left(InjkG\right)$ with diagonal $G$-action corresponds to the $E_{\infty }$ tensor product on $D_{dg}\left(C^{\ast }\left(BG;k\right)\right)$.

We show that $K\left(InjkG\right)$ can be regarded as a slight enlargement of the stable module category $StModkG$. It has better formal properties inasmuch as the ordinary cohomology ring $H^{\ast }\left(G,k\right)$ is better behaved than the Tate cohomology ring ${Ĥ}^{\ast }\left(G,k\right)$.

It is also better than the derived category $D\left(ModkG\right)$, because the compact objects in $K\left(InjkG\right)$ form a copy of the bounded derived category $D^b\left(modkG\right)$, whereas the compact objects in $D\left(ModkG\right)$ consist of just the perfect complexes.

Finally, we develop the theory of support varieties and homotopy colimits in $K\left(InjkG\right)$.