Lattice theory of torsion classes: Beyond $\tau$-tilting theory

TL;DR

This paper develops a lattice-theoretic framework for torsion classes over finite-dimensional algebras, revealing strong structural properties and connections to representation theory, including new insights into preprojective algebras and Cambrian lattices.

Contribution

It introduces a comprehensive lattice-theoretic approach to torsion classes, including brick labelling, congruences, and applications to preprojective algebras and Dynkin quivers, extending beyond $ au$-tilting theory.

Findings

$ ors A$ is a complete, bialgebraic, and completely semidistributive lattice.

The brick labelling provides a representation-theoretic interpretation of lattice congruences.

In type A, algebraic quotients of $ ors ext{Pi}$ are exactly its Hasse-regular lattice quotients.

Abstract

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set $\operatorname{\mathsf{tors}} A$ of torsion classes over a finite-dimensional algebra $A$. We show that $\operatorname{\mathsf{tors}} A$ is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of $\operatorname{\mathsf{tors}} A$. In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that $\operatorname{\mathsf{tors}} A$ is completely congruence uniform. When $I$ is a two-sided ideal of $A$, $\operatorname{\mathsf{tors}} (A/I)$ is a lattice quotient of $\operatorname{\mathsf{tors}} A$ which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of $\operatorname{\mathsf{tors}} A$ that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras $\Pi$, for which $\operatorname{\mathsf{tors}} \Pi$ is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between $\operatorname{\mathsf{tors}} k Q$ and the Cambrian lattice when $Q$ is a Dynkin quiver. We also prove that, in type $A$, the algebraic quotients of $\operatorname{\mathsf{tors}} \Pi$ are exactly its Hasse-regular lattice quotients.