Stratification and $$\pi $$ π -cosupport: finite groups

Abstract

Abstract. We introduce the notion of π-cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group, we use this to give a new proof of the classification of tensor ideal localising subcategories. In a sequel to this paper, we carry out the corresponding classification for finite group schemes.

Colocalising subcategories of modules over finite group schemes

“…Support and cosupport. The following definitions of π-support and π-cosupport of a G-module M are from [16] and [5] respectively. We set…”

“…As an example, we describe the point modules for the Klein four group, following the description of the π-points in [16, Example 2.3]; see also [5,Example 2.6]. are pairwise non-isomorphic; for example, their annihilators differ.…”

“…Let G be a finite group scheme over a field k of positive characteristic. There is a notion of π-cosupport [5] for any G-module M , based on the notion of π-points of G introduced by Friedlander and the fourth author [15]. The π-cosupport of M , denoted π-cosupp G (M ), is a subset of Proj H * (G, k).…”

Stratification for module categories of finite group schemes

“…This hinges on Chouinard's theorem that a G-module is projective if and only if its restriction to all elementary abelian subgroups of G is projective. Such a reduction is an essential step also in a second proof of the classification theorem for finite groups described in [11]. For general finite group schemes there is no straightforward replacement for a detecting family of subgroups akin to elementary abelian subgroups of finite groups: for any such family one needs to allow scalar extensions.…”

“…When p is a closed point in Proj H * (G, k), we verify this by using a new invariant of G-modules called π-cosupport introduced in [11], and recalled in Section 1. Its relevance to the problem on hand stems from the equality below; see Theorem 1.10.…”