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 $\mathrm{mod}\mathcal{A}$ denote the category of locally finite dimensional $\mathcal{A}$-modules, that is, the category of covariant functors $\mathcal{A}\to \mathrm{mod}k$. We prove that an irreducible monomorphism in $\mathrm{mod}\mathcal{A}$ has a finitely generated cokernel, and that an irreducible epimorphism in $\mathrm{mod}\mathcal{A}$ has a finitely co-generated kernel. Using this, we get that an almost split sequence in $\mathrm{mod}\mathcal{A}$ 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 $\mathrm{rep}(Q)$, the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of $\mathrm{rep}(Q)$.

arXiv comment17 pages, 1 figure