G-stable support $\tau$-tilting modules

TL;DR

This paper extends $ au$-tilting theory to incorporate group actions, establishing bijections among various $G$-stable modules, complexes, and torsion classes, and explores their relation to skew group algebras.

Contribution

It introduces $G$-stable support $ au$-tilting modules and proves bijections with $G$-stable complexes and torsion classes, linking to cluster-tilting objects and skew group algebras.

Findings

Established bijections among $G$-stable support $ au$-tilting modules, complexes, and torsion classes.

Connected $G$-stable support $ au$-tilting modules to $G$-stable cluster-tilting objects.

Analyzed the relationship between stable support $ au$-tilting modules and skew group algebras.

Abstract

Motivated by $\tau$-tilting theory developed by Adachi, Iyama and Reiten, for a finite-dimensional algebra $\Lambda$ with action by a finite group $G$, we introduce the notion of $G$-stable support $\tau$-tilting modules. Then we establish bijections among $G$-stable support $\tau$-tilting modules over $\Lambda$, $G$-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective $\Lambda$-modules, and $G$-stable functorially finite torsion classes in the category of finitely generated left $\Lambda$-modules. In the case when $\Lambda$ is the endomorphism of a $G$-stable cluster-tilting object $T$ over a Hom-finite 2-Calabi-Yau triangulated category $\mathcal{C}$ with a $G$-action, these are also in bijection with $G$-stable cluster-tilting objects in $\mathcal{C}$. Moreover, we investigate the relationship between stable support $\tau$-tilitng modules over $\Lambda$ and the skew group algebra $\Lambda G$.