A reduction theorem for $\tau$-rigid modules

TL;DR

This paper establishes a reduction theorem linking support τ-tilting modules over an algebra and its quotient by a specific ideal, enabling classification of these modules for various algebra types, including group ring blocks.

Contribution

It introduces an explicit bijection reducing τ-tilting module classification to string algebras, and applies this to classify modules over tame blocks of group rings.

Findings

Existence of τ-tilting finite wild blocks with multiple simple modules

Complete classification of support τ-tilting modules for certain algebra classes

Bound of 32 basic two-term tilting complexes for tame blocks

Abstract

We prove a theorem which gives a bijection between the support $\tau$-tilting modules over a given finite-dimensional algebra $A$ and the support $\tau$-tilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the center of $A$ and the radical of $A$. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are $\tau$-tilting finite wild blocks with more than one simple module. We then go on to classify all support $\tau$-tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most $32$ different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field $k$, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all $\tau$-rigid modules over (not necessarily symmetric) string algebras.