Some relative stable categories are compactly generated

TL;DR

This paper investigates the conditions under which the relative stable module category of a finite group is compactly generated, establishing that it occurs when the subgroup's group algebra has finite representation type, notably for cyclic Sylow p-subgroups.

Contribution

It proves that the relative stable category is compactly generated if the subgroup's group algebra has finite representation type, a new insight linking algebraic properties to category generation.

Findings

Relative stable category is compactly generated when H has finite representation type.

Finite representation type of H is equivalent to Sylow p-subgroups of H being cyclic.

Provides new conditions for compact generation in relative stable categories.

Abstract

Let G be a finite group. The stable module category of G has been applied extensively in group representation theory. In particular, it has been used to great effect that it is a triangulated category which is compactly generated. Let H be a subgroup of G. It is possible to define a stable module category of G relative to H. It too is a triangulated category, but no non-trivial examples have been known where this relative stable category was compactly generated. We show here that the relative stable category is compactly generated if the group algebra of H has finite representation type. In characteristic p, this is equivalent to the Sylow p-subgroups of H being cyclic.