Vol. 2, No. 1, 2008

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Complexes of injective kG-modules

David John Benson and Henning Krause

Vol. 2 (2008), No. 1, 1–30
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 $K\left(InjkG\right)$.

Keywords
modular representation theory, derived category, stable module category, cohomology of group
Primary: 20C20
Secondary: 20J06