Locally noetherian quiver representations

TL;DR

This paper characterizes when quiver representations are locally noetherian, linking the property to the quiver's structure and extending the result to categories with Gr"obner enrichments.

Contribution

It establishes a characterization of left noetherian quivers and generalizes the result to small categories with Gr"obner enrichments.

Findings

A quiver is left noetherian iff its path ideals satisfy ACC.

The category of quiver representations is locally noetherian under these conditions.

The proof extends to small categories with Gr"obner enrichments.

Abstract

It is shown that a quiver is left noetherian if and only if the category of quiver representations in any locally noetherian abelian category is again locally noetherian. Here, locally noetherian means that any object is the directed union of its noetherian subobjects. For a quiver to be left noetherian means that the left ideals of paths starting at any fixed vertex satisfy the ascending chain condition. The proof generalises to representations of any small category that admits a Gr\"obner enrichment.