Colocalising subcategories of modules over finite group schemes

Abstract

The Hom closed colocalising subcategories of the stable module category of a finite group scheme are classified. This complements the classification of the tensor closed localising subcategories in our previous work. Both classifications involve pi-points in the sense of Friedlander and Pevtsova. We identify for each pi-point an endofinite module which both generates the corresponding minimal localising subcategory and cogenerates the corresponding minimal colocalising subcategory.Comment: 17 pages, final version to appear in Annals of K-Theory. The duality statement in Theorem 3.1 of v1 has been removed since it is incorrect, and some subsequent arguments were modifie

The variety of subadditive functions for finite group schemes

“…In this section we recall some of the results from our joint work with Iyengar and Pevtsova [2,3]. The first result is a classification of tensor closed thick subcategories of mod G that has been anticipated in [6].…”

“…To each π-point α : K[t]/(t p ) → KG corresponds a point module ∆ G (α) := res K k (Hom K[t]/(t p ) (KG, K)). This is an endofinite G-module and plays a prominent role in recent work with Iyengar and Pevtsova [3]. which restricts to submodules over the endomorphisms rings of ∆ G (α) and K respectively.…”

“…We examine this correspondence in the context of finite group schemes, for certain endofinite 'point modules' for a finite group scheme G introduced in our joint work with Iyengar and Pevtsova [3]. These point modules come from the π-points introduced by Friedlander and Pevtsova [6].…”

Local duality for representations of finite group schemes

“…For a field extension extension of scalars and restriction give exact functors The functors form an adjoint pair, with the left adjoint mapping to and respecting tensor products, so one has a well-known projection formula: for a -module and -module ; see [BDS16, (2.16)] or [BIKP17, Lemma 2.2].…”

Stratification for module categories of finite group schemes

“…The methods developed in this work have led to other new results concerning the structure of the stable module category of a finite group scheme, including a classification of its Hom closed colocalising subcategories [10], and to a type of local Serre duality theorem for StMod G; see [12].…”