A non-existence theorem for almost split sequences

TL;DR

This paper proves that for certain infinite quivers, almost split sequences only end at finitely presented, non-projective indecomposable representations, clarifying the structure of these sequences in such contexts.

Contribution

It establishes a non-existence theorem for almost split sequences ending at non-finitely presented indecomposables in bound quivers with countably many vertices.

Findings

Almost split sequences do not end at non-finitely presented indecomposables.

Finitely presented, non-projective indecomposables are exactly the ending terms of almost split sequences.

Dual results hold for the starting terms of almost split sequences.

Abstract

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I has finite dimensional Hom-spaces. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q,I). We then get that an indecomposable representation M of (Q,I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.