{\nu}-stable {\tau}-tilting modules

TL;DR

This paper introduces nu-stable support tau-tilting modules for selfinjective algebras, establishing bijections with various tilting and torsion structures, and explores their relation to 2-CY tilted algebras.

Contribution

It defines nu-stable support tau-tilting modules and establishes bijections with tilting complexes, torsion classes, and cluster tilting objects, extending tau-tilting theory.

Findings

Bijections between nu-stable support tau-tilting modules and tilting complexes

Correspondence with functorially finite torsion classes

Necessary conditions for selfinjective algebras to be 2-CY tilted

Abstract

Inspired by tau-tilting theory [AIR], we introduce the notion of nu-stable support tau-tilting modules. For any finite dimensional selfinjective algebra {\Lambda}, we give bijections between two-term tilting complexes in Kb(proj {\Lambda}), nu-stable support tau-tilting {\Lambda}-modules and nu-stable functorially finite torsion classes in mod {\Lambda}. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in C if {\Lambda} is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category C. As an application, we give a necessary condition such that selfinjective algebras are 2-CY tilted.