COMPLEXES OF INJECTIVE kG-MODULES

Let G be a finite group and k be a field of characteristic p. We investigate the homotopy category K(Inj kG) of the category C(Inj kG) 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(Inj kG) with diagonal G-action corresponds to the E∞ tensor product on Ddg(C(BG; k)). We show that K(Inj kG) can be regarded as a slight enlargement of the stable module category StMod kG. 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(Mod kG), because the compact objects in K(Inj kG) form a copy of the bounded derived category D(mod kG), whereas the compact objects in D(Mod kG) consist of just the perfect complexes. Finally, we develop the theory of support varieties and homotopy colimits in K(Inj kG).