Mutation of torsion pairs for finite-dimensional algebras

TL;DR

This paper explores the mutation of torsion pairs in the module category of finite-dimensional algebras, linking lattice structures, rigid sets, and silting mutation to better understand their combinatorial and topological properties.

Contribution

It provides an explicit description of mutation operations on torsion pairs and relates these to rigid sets in the Ziegler spectrum, extending previous results to include non-mutable points.

Findings

Established a bijection between wide intervals and closed rigid sets.

Connected arrows in the Hasse quiver to almost complete rigid sets.

Generalized existing results by Adachi, Iyama, and Reiten.

Abstract

We study the lattice $\mathbf{tors}(A)$ of torsion pairs in the category $\mathrm{mod}(A)$ of finitely generated modules over an artinian ring $A$. It was shown by the authors in previous work that $\mathbf{tors}(A)$ is isomorphic to a lattice formed by certain closed sets, called maximal rigid, in the Ziegler spectrum of the unbounded derived category $\mathrm{D}(A)$ of $A$. Moreover, the structure of this lattice is described by an operation on maximal rigid sets which encompasses (the dual of) silting mutation. In this paper we provide an explicit description of this operation and we discuss how it is reflected in the lattice $\mathbf{tors}(A)$. We establish a bijection between the wide intervals in $\mathbf{tors}(A)$ and the closed rigid sets in the Ziegler spectrum of $\mathrm{D}(A)$. Moreover, we show that the arrows in the Hasse quiver of $\mathbf{tors}(A)$ correspond to the closed rigid sets that are almost complete, or equivalently, that can be completed to a maximal rigid set in exactly two ways. Our results are most interesting in the case when $A$ is a finite dimensional algebra. In fact, we generalise results by Adachi, Iyama and Reiten, with an important difference: not every point in a maximal rigid set is mutable. We use the topology on the Ziegler spectrum to determine the mutable points. In the last section of the paper we illustrate our results by the example of a finite dimensional algebra arising from a triangulation of an annulus.