\tau-rigid modules over Auslander algebras

TL;DR

This paper characterizes τ-rigid modules over Auslander algebras using projective dimension and establishes conditions under which the algebra is τ-tilting finite, linking module properties to algebra finiteness.

Contribution

It provides a new characterization of τ-rigid modules over Auslander algebras and identifies conditions for τ-tilting finiteness based on module grades and projective dimensions.

Findings

Characterization of τ-rigid modules via projective dimension.

Conditions for τ-tilting finiteness in Auslander algebras.

Relationship between module grade and τ-tilting finiteness.

Abstract

We give a characterization of $\tau$-rigid modules over Auslander algebras in terms of projective dimension of modules. Moreover, we show that for an Auslander algebra $\Lambda$ admitting finite number of non-isomorphic basic tilting $\Lambda$-modules and tilting $\Lambda^{op}$-modules, if all indecomposable $\tau$-rigid $\Lambda$-modules of projective dimension $2$ are of grade $2$, then $\Lambda$ is $\tau$-tilting finite.