Lattices of pretorsion classes

TL;DR

This paper introduces and analyzes the lattice of pretorsion classes in module categories over finite-dimensional algebras, revealing their structure, irreducible elements, and conditions for distributivity, and explores their applications in constructing pretorsion theories.

Contribution

It provides a comprehensive description of the lattice of pretorsion classes, characterizes its irreducible elements, and establishes conditions for distributivity and connections to torsion class lattices.

Findings

Fully describes the join-irreducible elements of the lattice.

Characterizes when the lattice is distributive.

Shows how to construct pretorsion theories from these lattices.

Abstract

Since their introduction, torsion theories have played a key role in the study of abelian and pointed categories. In representation theory, torsion theories and lattices of torsion classes of mod$ A$, for $A$ a finite-dimensional algebra, have been widely studied. The more recent definition of pretorsion theories, that can be given for any category, has expanded the theory, giving many more instances of ``non-pointed torsion theories'' in unexpected settings. In this work, we introduce and study the lattice $\mathcal{L}_t(A)$ of pretorsion classes of mod$ A$. These lattices are in close connection with the lattices tors$ A$ of torsion classes of mod$ A$. We fully describe the completely join-irreducible elements of $\mathcal{L}_t(A)$. Moreover, we characterise and give a full classification of when $\mathcal{L}_t(A)$ is distributive and further describe when it can be identified with the \emph{distributive closure} of tors$ A$. Finally, we show how the lattices of pretorsion classes, together with their duals, can be used to build pretorsion theories in mod$ A$.