Torsion pairs via cosilting mutation

TL;DR

This work develops a large-silting perspective on the lattice of torsion pairs $\mathbf{tors}(A)$ for a finite dimensional algebra $A$ by connecting it to cosilting mutation and the Ziegler spectrum. It proves a bijection between $\mathbf{2}\mathbf{-Cosilt}(A)$ and $\mathbf{f{-}Tors}(A)$ and, for left artinian $A$, a lattice isomorphism $\mathbf{tors}(A) \cong \mathbf{MaxRigid}(A)$, with maximal rigid sets encoding torsion data via $\sigma$-derived objects and neg-isolated points. Mutation of cosilting complexes corresponds to exchanging neg-isolated points inside maximal rigid sets, while wide intervals in $\mathbf{tors}(A)$ correspond to closed rigid sets in the Ziegler spectrum, providing a geometric/topological lens on mutations. The paper also delivers three applications to (i) bricks and grains, (ii) brick-finite versus brick-infinite algebras, and (iii) semistable torsion pairs, thereby unifying torsion theory, cosilting theory, and representation-theoretic invariants in a coherent framework.

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.

Theorems & Definitions (32)

• Theorem 2.1
• Remark 2.2
• Proposition 3.1
• Lemma 3.2
• Theorem 3.3
• Proposition 4.1
• Theorem 4.2
• Corollary 4.3
• proof
• Remark 4.4