Reduction theorem for support $\tau$-tilting modules over group algebras

TL;DR

This paper extends the reduction techniques for classifying support τ-tilting modules over group algebra blocks, demonstrating that quotient reduction is valid under certain conditions, thus aiding the understanding of derived categories.

Contribution

It introduces a quotient reduction method for support τ-tilting modules, complementing existing subgroup reduction techniques in the context of group algebra blocks.

Findings

Support τ-tilting module classification can be reduced via quotient groups.

Quotient reduction is valid under specific algebraic conditions.

The method simplifies the study of derived categories of group algebras.

Abstract

In studying the structure of derived categories of module categories of group algebras or their blocks, it is fundamental to classify support $\tau$-tilting modules. Koshio and Kozakai showed that the structure of support $\tau$-tilting modules over blocks of finite groups can be reduced to that of their subgroups under suitable conditions. We show that the 'quotient reduction' is also valid under suitable conditions.