Mutation and torsion pairs

TL;DR

This paper develops a generalized mutation theory for silting and cosilting t-structures in triangulated categories, linking it to torsion class inclusions and extending classical mutation concepts.

Contribution

It introduces a broad mutation framework for silting and cosilting objects, connecting it to torsion class inclusions and classical mutation theory in representation theory.

Findings

Mutation of pure-injective cosilting objects generalizes classical silting mutations.

Minimal torsion class inclusions correspond to irreducible mutations.

Extends mutation concepts to broader triangulated category frameworks.

Abstract

Mutation of compact silting objects is a fundamental operation in the representation theory of finite-dimensional algebras due to its connections to cluster theory and to the lattice of torsion pairs in module or derived categories. In this paper we develop a theory of mutation in the broader framework of silting or cosilting t-structures in triangulated categories. We show that mutation of pure-injective cosilting objects encompasses the classical concept of mutation for compact silting complexes. As an application we prove that any minimal inclusion of torsion classes in the category of finitely generated modules over an artinian ring corresponds to an irreducible mutation. This generalises a well-known result for functorially finite torsion classes.