Local subgroups and the stable category

Abstract:

If $G$ is a finite group and $k$ is an algebraically closed field of characteristic $p>0$, then this paper uses the local subgroup structure of $G$ to define a category $\mathfrak {L}(G,k)$ that is equivalent to the stable category of all left $kG$-modules modulo projectives. A subcategory of $\mathfrak {L}(G,k)$ equivalent to the stable category of finitely generated $kG$-modules is also identified. The definition of $\mathfrak {L}(G,k)$ depends largely but not exclusively upon local data; one condition on the objects involves compatibility with respect to conjugations by arbitrary group elements rather than just elements of $p$-local subgroups.