Torsion pairs via cosilting mutation

TL;DR

This paper explores the structure of torsion pairs in finitely generated modules over a left artinian ring, linking them to cosilting mutations and topological closed sets in the Ziegler spectrum, with applications to finite-dimensional algebras.

Contribution

It introduces a novel connection between torsion pairs and cosilting mutations via topological closed sets in the Ziegler spectrum, providing new insights into their lattice structure.

Findings

Torsion pairs are adjacent iff related by cosilting mutation.

Describes the mutation operation from multiple perspectives.

Applies the theory to finite-dimensional algebras.

Abstract

For a left artinian ring A, we study the lattice torsA of torsion pairs in the category of finitely generated A-modules by considering an isomorphic lattice formed by certain closed sets in a topological space associated to A, the Ziegler spectrum of the unbounded derived category of ModA. Torsion pairs in torsA turn out to be adjacent if and only if the associated closed sets are related by an operation which is induced by mutation of cosilting complexes. We describe this operation from several perspectives and present a number of applications in the case when A is a finite dimensional algebra.