Irreducible morphisms and locally finite dimensional representations

TL;DR

This paper investigates the structure of irreducible morphisms in locally finite dimensional module categories over certain additive categories, revealing finiteness properties and describing the shapes of Auslander-Reiten quivers for representations of strongly locally finite quivers.

Contribution

It establishes finiteness conditions for irreducible monomorphisms and epimorphisms, and characterizes the structure of almost split sequences and Auslander-Reiten quivers in this context.

Findings

Irreducible monomorphisms have finitely generated cokernels.

Irreducible epimorphisms have finitely co-generated kernels.

The shapes of Auslander-Reiten quivers for strongly locally finite quivers are classified.

Abstract

Let $\mathcal{A}$ be a Hom-finite additive Krull-Schmidt $k$-category where $k$ is an algebraically closed field. Let ${\rm mod} \mathcal{A}$ denote the category of locally finite dimensional $\mathcal{A}$-modules, that is, the category of covariant functors $\mathcal{A} \to {\rm mod} k$. We prove that an irreducible monomorphism in ${\rm mod} \mathcal{A}$ has a finitely generated cokernel, and that an irreducible epimorphism in ${\rm mod} \mathcal{A}$ has a finitely co-generated kernel. Using this, we get that an almost split sequence in ${\rm mod} \mathcal{A}$ has to start with a finitely co-presented module and end with a finitely presented one. Finally, we apply our results in the study of ${\rm rep}(Q)$, the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of ${\rm rep}(Q)$.