Complexes of injective \(kG\)-modules. (Q1005852)

Let \(k\) be a field of characteristic \(p\) and let \(G\) be a finite group. Let \(C(\text{Inj\,}kG)\) be the category of chain complexes of injective \(k\)G-modules, and let \(K(\text{Inj\,}kG)\) be the corresponding homotopy category. The authors investigate a recollement relating \(K(\text{Inj\,}kG)\) to the stable module category \(\text{StMod\,}kG\) and the derived category \(D(\text{Mod\,}kG)\): \[ \text{StMod\,}kG{\underset \longleftarrow {\overset \longleftarrow \longrightarrow}} K(\text{Inj\,}kG) {\underset \longleftarrow {\overset \longleftarrow \longrightarrow}} D(\text{Mod\,}kG). \] The compact objects in these categories are \(\text{stmod\,}kG\leftarrow D^b(\text{mod\,}kG)\leftarrow D^b(\text{proj\,}kG)\). This means that \(K(\text{Inj\,}kG)\) can be seen as the ``big'' category for \(D^b(\text{mod\,}kG)\), whereas \(D(\text{Mod\,}kG)\) does not have enough compact objects. The category \(K(\text{Inj\,}kG)\) provides an algebraic replacement for the derived category of the differential graded algebra of singular cochains on the classifying space, \(D_{dg}(C^*(BG;k))\). Namely, if \(G\) is a \(p\)-group, there is an equivalence of categories \(K(\text{Inj\,}kG)\cong D_{dg}(C^*(BG;k))\). The authors show that the tensor product over \(k\) of complexes in \(K(\text{Inj\,}kG)\) corresponds under this equivalence to the \(E_\infty\) tensor product on \(D_{dg}(C^*(BG;k))\). If \(G\) is not a \(p\)-group, the authors obtain the following result. Writing \(ik\) for an injective resolution of the trivial \(kG\)-module \(k\), they show that there is an equivalence between \(D_{dg}(C^*(BG;k))\) and the localizing subcategory of \(K(\text{Inj\,}kG)\) generated by \(ik\). The authors also develop the theory of support varieties for objects in \(K(\text{Inj\,}kG)\), extending the theory developed by \textit{D. J. Benson, J. F. Carlson} and \textit{J. Rickard} [Math. Proc. Camb. Philos. Soc. 120, No. 4, 597-615 (1996; Zbl 0888.20003)].