Lattices of pretorsion classes

TL;DR

We study lattices of pretorsion classes $\mathcal{L}_t(A)$ in $\mathsf{mod} A$ for finite-dimensional $k$-algebras, clarifying how they extend the classical torsion-class lattice $\mathsf{tors} A$ by including all indecomposables via $\mathrm{Gen}(M)$. A core result is that indecomposable modules index the completely join-irreducible elements of $\mathcal{L}_t(A)$ through $M\mapsto \mathrm{Gen}(M)$, and that $\mathcal{L}_t(A)$ is distributive precisely when every indecomposable has a unique maximal submodule (with a full bound-quiver classification). The distributive closure of $\mathsf{tors} A$ can be identified with $\mathcal{L}_t(A)$ in the locally representation directed, finite-type/string-algebra setting, tying pretorsion structures to classical torsion theory through join-irreducibles. The paper also provides a dual lattice of pretorsion-free classes, methods to build pretorsion theories from the lattices, and an alternative realisation of $\mathcal{L}_t(A)$ as order-ideals of its join-irreducibles, enabling explicit computations and illustrations in low-dimensional examples.

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$.

Figures (5)

• Figure 1: Inverse string (left), Direct string (middle), other string (right).
• Figure 3: On the left, the lattice of pretorsion classes of $\mathsf{mod} k\mathbb{A}_2$, and on the right its lattice of pretorsion-free classes. In both cases, the join-irreducible elements are coloured in red. Dotted frames indicate the elements of the torsion and torsion-free classes.
• Figure 4: The distributive lattice ${\mathcal{L}}_t(kQ)$ for $Q: 1\rightarrow 2\rightarrow 3$.
• Figure 5: The non-distributive lattice ${\mathcal{L}}_t(kQ)$ for $Q: 1\rightarrow 2\leftarrow 3$.
• Figure 6: The distributive lattice ${\mathcal{L}}_t(kQ)$ for $Q: 1\leftarrow 2\rightarrow 3$.

Theorems & Definitions (84)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Definition 2.7
• Proposition 2.8
• proof