Irreducible Morphisms and Locally Finite Dimensional Representations

Abstract

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