Direct Limit closure of induced Quiver Representations

TL;DR

This paper generalizes the characterization of flat and Gorenstein flat quiver representations from specific cases to a broad categorical setting, providing new insights into their structure and properties.

Contribution

It extends the filtering results of quiver representations to any class in AB5-abelian categories and characterizes Gorenstein flat representations over right coherent rings.

Findings

Generalized filtering of $ ext{Phi}( ext{X})$ for any class in AB5 categories

Characterized Gorenstein flat quiver representations over right coherent rings

Provided a categorical framework for flat and Gorenstein flat representations

Abstract

In 2004 and 2005 Enochs et al. characterized the flat and projective quiver-representations of left rooted quivers. The proofs can be understood as filtering the classes $\Phi(\operatorname{Add}\mathscr X)$ and $\Phi(\varinjlim\mathscr X)$ when $\mathscr X$ is the finitely generated projective modules over a ring. In this paper we generalize the above and show that $\Phi(\mathscr X)$ can always be filtered for any class $\mathscr X$ in any AB5-abelian category. With an emphasis on $\Phi(\varinjlim\mathscr X)$ we investigate the Gorenstein homological situation. Using an abstract version of Pontryagin duals in abelian categories we give a more general characterization of the flat representations and end up by describing the Gorenstein flat quiver representations over right coherent rings.