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Mutation of torsion pairs for finite-dimensional algebras

Lidia Angeleri Hügel, Rosanna Laking, Francesco Sentieri

TL;DR

This work extends the Adachi–Iyama–Reiten cosilting framework to left artinian (and in particular finite-dimensional) algebras by identifying the lattice of torsion pairs with closed maximal rigid sets in the Ziegler spectrum of the derived category D(A). It develops mutation of maximal rigid sets via HRS-tilts and width-enlargement, and establishes a precise correspondence between wide intervals in tors(A) and closed rigid subsets in Zg(D(A)); irreducible mutations are characterized topologically in terms of almost-complete rigid sets and two-completion phenomena. The paper also connects mutability to the topology of the Ziegler spectrum (neg-isolated and simple objects in hearts) and provides a detailed, geometry-driven example for cluster-tilted algebras of type tilde A arising from annulus triangulations. Overall, it reveals strong structural parallels between silting/cosilting mutation and torsion-theoretic lattices in a broader artinian setting, while highlighting new phenomena (non-all-points-mutable) governed by purity-topology. Practical implications include a unified view of mutation in module categories and derived categories, and explicit combinatorial models via Ziegler-geometry for notable algebras.

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.

Mutation of torsion pairs for finite-dimensional algebras

TL;DR

This work extends the Adachi–Iyama–Reiten cosilting framework to left artinian (and in particular finite-dimensional) algebras by identifying the lattice of torsion pairs with closed maximal rigid sets in the Ziegler spectrum of the derived category D(A). It develops mutation of maximal rigid sets via HRS-tilts and width-enlargement, and establishes a precise correspondence between wide intervals in tors(A) and closed rigid subsets in Zg(D(A)); irreducible mutations are characterized topologically in terms of almost-complete rigid sets and two-completion phenomena. The paper also connects mutability to the topology of the Ziegler spectrum (neg-isolated and simple objects in hearts) and provides a detailed, geometry-driven example for cluster-tilted algebras of type tilde A arising from annulus triangulations. Overall, it reveals strong structural parallels between silting/cosilting mutation and torsion-theoretic lattices in a broader artinian setting, while highlighting new phenomena (non-all-points-mutable) governed by purity-topology. Practical implications include a unified view of mutation in module categories and derived categories, and explicit combinatorial models via Ziegler-geometry for notable algebras.

Abstract

We study the lattice of torsion pairs in the category of finitely generated modules over an artinian ring . It was shown by the authors in previous work that is isomorphic to a lattice formed by certain closed sets, called maximal rigid, in the Ziegler spectrum of the unbounded derived category of . 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 . We establish a bijection between the wide intervals in and the closed rigid sets in the Ziegler spectrum of . Moreover, we show that the arrows in the Hasse quiver of 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 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.
Paper Structure (22 sections, 47 theorems, 31 equations, 7 figures)

This paper contains 22 sections, 47 theorems, 31 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The setup
  4. Notation
  5. The cosilting bijections
  6. Relationship with cosilting modules and cosilting complexes
  7. Cosilting and t-structures
  8. Cosilting and the Ziegler spectrum
  9. Mutation of maximal rigid sets
  10. Mutation and HRS-tilting
  11. Mutation and wide intervals
  12. Mutation and simple objects
  13. Critical and special objects
  14. Wide intervals and closed rigid subsets
  15. Topological characterisations of irreducible mutation
  16. ...and 7 more sections

Key Result

Lemma 2.2

ALS1 Let $\sigma,\mu \in \mathrm{K}^{[0,1]}(\mathrm{Inj}A)$ and $M,N\in\mathrm{Mod}(A)$.

Figures (7)

  • Figure 1: Top left: An annulus with 3 marked points on the inner boundary and 4 marked points on the outer boundary. Top right: The universal cover of the same annulus with marked points. Bottom left: Example of a triangulation with arrows of quiver indicated. Bottom middle: The same triangulation in the universal cover. Bottom right: Quiver with relations determined by the triangulation.
  • Figure 2: Left: Example of an asymptotic are $\alpha$ and a finite arc $\beta$. Right: The same arcs $\alpha$ and $\beta$ occurring in the universal cover.
  • Figure 3: The unique closed curve up to homotopy representing the family of band modules.
  • Figure 4: An example of a set $\mathcal{C}$ that satisfies the conditions with respect to $\Gamma$ defined in Figure \ref{['fig:quiver']} but does not containing any arc that crosses $\varepsilon$. We note that $\gamma_2$ is contained in $\Gamma$.
  • Figure 5: An example of a set $\mathcal{C}$ that satisfies the conditions with respect to $\Gamma$ defined in Figure \ref{['fig:quiver']} and contains an arc that crosses $\varepsilon$. We note that $\gamma_3$ and $\gamma_7$ are contained in $\Gamma$.
  • ...and 2 more figures

Theorems & Definitions (105)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.6
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • Proposition 2.11
  • ...and 95 more