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

Let G be a finite group and k be a field of characteristic p. We investigate the homotopy category K(InjkG) of the category C(InjkG) 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 K(InjkG) with diagonal G-action corresponds to the E tensor product on Ddg(C(BG;k)).

We show that K(InjkG) 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(G,k) is better behaved than the Tate cohomology ring Ĥ(G,k).

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

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

modular representation theory, derived category, stable module category, cohomology of group
Mathematical Subject Classification 2000
Primary: 20C20
Secondary: 20J06
Received: 1 February 2007
Revised: 22 November 2007
Accepted: 24 December 2007
Published: 1 February 2008
David John Benson
Department of Mathematics
University of Aberdeen
Aberdeen AB24 3EA
United Kingdom
Henning Krause
Institut für Mathematik
Universität Paderborn
33095 Paderborn