Abstract

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 D_dg(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 D_dg(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 D^b(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).

Keywords

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

Mathematical Subject Classification

Primary: 20C20

Secondary: 20J06

Authors