Complexes of injective kG-modules

Benson, David; Krause, Henning

doi:10.2140/ant.2008.2.1

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Complexes of injective kG-modules

David John Benson and Henning Krause

Vol. 2 (2008), No. 1, 1–30

DOI: 10.2140/ant.2008.2.1

Abstract

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

We show that $K (I n j k G)$ can be regarded as a slight enlargement of the stable module category $S t M o d k G$ . It has better formal properties inasmuch as the ordinary cohomology ring $H^{*} (G, k)$ is better behaved than the Tate cohomology ring $Ĥ^{*} (G, k)$ .

It is also better than the derived category $D (M o d k G)$ , because the compact objects in $K (I n j k G)$ form a copy of the bounded derived category $D^{b} (m o d k G)$ , whereas the compact objects in $D (M o d k G)$ consist of just the perfect complexes.

Finally, we develop the theory of support varieties and homotopy colimits in $K (I n j k G)$ .

Keywords

modular representation theory, derived category, stable module category, cohomology of group

Mathematical Subject Classification 2000

Primary: 20C20

Secondary: 20J06

Milestones

Received: 1 February 2007

Revised: 22 November 2007

Accepted: 24 December 2007

Published: 1 February 2008

Authors

David John Benson
Department of Mathematics University of Aberdeen Aberdeen AB24 3EA Scotland United Kingdom

Henning Krause
Institut für Mathematik Universität Paderborn 33095 Paderborn Germany

Vol. 2, No. 1, 2008