Projective objects in the category of pointwise finite dimensional representations of an interval finite quiver

TL;DR

This paper classifies indecomposable projective objects in the category of pointwise finite dimensional representations of interval finite quivers, showing they are direct sums of countably generated flat representations, advancing understanding of their structure.

Contribution

It introduces a class of flat representations and provides a complete classification of indecomposable projective objects in this category.

Findings

Indecomposable projectives are characterized as direct sums of countably generated flat representations.

A new class of flat representations is introduced for interval finite quivers.

The classification enhances understanding of the structure of pointwise finite dimensional representations.

Abstract

For an interval finite quiver $Q$, we introduce a class of flat representations. We classify the indecomposable projective objects in the category $\mathrm{rep}(Q)$ of pointwise finite dimensional representations. We show that an object in $\mathrm{rep}(Q)$ is projective if and only if it is a direct sum of countably generated flat representations.