Support varieties for transporter category algebras

TL;DR

This paper develops a support variety theory for transporter category algebras, extending existing theories from group cohomology to equivariant cohomology, and establishes a Quillen stratification for modules.

Contribution

It introduces a novel support variety framework for transporter category algebras, generalizing Carlson's theory and applying Snashall-Solberg's support varieties to new algebra classes.

Findings

Supports a Quillen stratification for modules

Extends Carlson's support variety theory to equivariant cohomology

Provides a framework for non selfinjective algebras

Abstract

Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G-space BP, while the k-category algebra of the transporter category is the (Gorenstein) skew group algebra on the G-incidence algebra kP. We introduce a support variety theory for the category algebras of transporter categories. It extends Carlson's support variety theory on group cohomology rings to equivariant cohomology rings. In the mean time it provides a class of (usually non selfinjective) algebras to which Snashall-Solberg's (Hochschild) support variety theory applies. Various properties will be developed. Particularly we establish a Quillen stratification for modules.