τ-rigid modules for algebras with radical square zero

Abstract

In this paper, we show that for an algebra $Λ$ with radical square zero and an indecomposable $Λ$-module $M$ such that $Λ$ is Gorenstein of finite type or $τM$ is $τ$-rigid, $M$ is $τ$-rigid if and only if the first two projective terms of a minimal projective resolution of $M$ have no on-zero direct summands in common. We also determined all $τ$-tilting modules for Nakayama algebras with radical square zero. Moreover, by giving a construction theorem we show that a basic connected radical square zero algebra admitting a unique $τ$-tilting module is local.