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

Abstract

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set $\mathrm{tors}A$ of torsion classes over a finite-dimensional algebra $A$. We show that $\mathrm{tors}A$ 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 $\mathrm{tors}A$. In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that $\mathrm{tors}A$ is completely congruence uniform. When $I$ is a two-sided ideal of $A$, $\mathrm{tors}(A/I)$ is a lattice quotient of $\mathrm{tors}A$ 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 $\mathrm{tors}A$ 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 $\mathrm{tors}\Pi$ is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between $\mathrm{tors}kQ$ and the Cambrian lattice when $Q$ is a Dynkin quiver. We also prove that, in type $A$, the algebraic quotients of $\mathrm{tors}\Pi$ are exactly its Hasse-regular lattice quotients.

arXiv commentv2: Many improvements compared to the first version (in particular, more discussion about complete congruence uniform lattices). v3: Added subtitle and made other minor changes. v4: Post-publication correction of an error in the proof of Proposition 2.6. (Thanks to Lorenzo Molena for pointing out the error.)

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• Celebrating Loday's Associahedron
• $\tau$-perpendicular wide subcategories
• The category of a partitioned fan
• On $\tau$-tilting finiteness of symmetric algebras of polynomial growth
• Mutation and torsion pairs
• Morphisms and extensions between bricks over preprojective algebras of type A
• Geometric realizations of the $s$-weak order and its lattice quotients
• A uniqueness property of τ exceptional sequences
• Relative left Bongartz completions and their compatibility with mutations
• $c$-Birkhoff polytopes
• $\tau$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras
• Separating trees and simple congruences of the weak order