Relative homological algebra in categories of representations of infinite quivers

TL;DR

This paper develops new methods in relative homological algebra for categories of quiver representations, establishing the existence of torsion free covers and componentwise flat covers under broad conditions.

Contribution

It proves the existence of torsion free covers and componentwise flat covers in categories of quiver representations, extending homological algebra techniques to infinite quivers.

Findings

Existence of torsion free covers in certain quiver representation categories.

Existence of componentwise flat covers and envelopes for all quivers and rings.

Applicable to a wide class of source injective representation quivers.

Abstract

In the first part of this paper, we prove the existence of torsion free covers in the category of representations of quivers, $(Q,R-Mod)$, for a wide class of quivers included in the class of the so-called source injective representation quivers provided that any direct sum of torsion free and injective $R$-modules is injective. In the second part, we prove the existence of $\mathscr{F}_{cw}$-covers and $\mathscr{F}_{cw}^{\perp}$-envelopes for any quiver $Q$ and any ring $R$ with unity, where $\mathscr{F}_{cw}$ is the class of all "componentwise" flat representations of $Q$.