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This research explores how descent to finite subgroups occurs in specific module categories related to discrete groups.

June 5, 2026Open Access

Descent and finite permutation resolutions for discrete groups

Key Points

This research explores how descent to finite subgroups occurs in specific module categories related to discrete groups.
Investigated the homotopy category of projective kG-modules and stable module categories.
Applied results regarding $FP_ ext{infty}$ modules and finite resolutions of permutation modules. Never examined under a specific sample size or subgroup.
Examined derived categories of permutation kG-modules with finite isotropy.
Demonstrated that the homotopy and stable module categories admit descent to finite subgroups.
Established that any $kG$-module of type $FP_ ext{infty}$ can be retracted to a finite resolution by finitely generated permutation modules.

Abstract

Let G be a discrete group with a finite-dimensional model for the classifying space for proper actions, and let k be a commutative Noetherian ring of finite global dimension. In this setting, we prove that the homotopy category of projective kG-modules, the stable module category of kG-modules as defined by Mazza-Symonds, and the derived category of permutation kG-modules with finite isotropy, admit descent to finite subgroups. As an application, we show that any kG-module of type FP_ is a retract of a module that admits a finite resolution by finitely generated -permutation modules with finite isotropy, generalizing a result of Balmer-Gallauer.

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