Gorenstein flat representations of left rooted quivers

TL;DR

This paper characterizes Gorenstein flat representations of left rooted quivers over arbitrary rings and establishes a model structure where these representations are the cofibrant objects.

Contribution

It provides a new characterization of Gorenstein flat representations in terms of module properties and constructs a hereditary abelian model structure on the category of quiver representations.

Findings

Gorenstein flat representations are characterized by injectivity of certain homomorphisms and Gorenstein flatness of modules involved.

A hereditary abelian model structure on the representation category is established with Gorenstein flat representations as cofibrant objects.

The model structure's fibrant and trivial objects are explicitly described in terms of cotorsion and orthogonal categories.

Abstract

We study Gorenstein flat objects in the category ${\sf Rep}(Q,R)$ of representations of a left rooted quiver $Q$ with values in ${\sf Mod}(R)$, the category of all left $R$-modules, where $R$ is an arbitrary associative ring. We show that a representation $X$ in ${\sf Rep}(Q,R)$ is Gorenstein flat if and only if for each vertex $i$ the canonical homomorphism $\varphi_i^X: \oplus_{a:j\to i}X(j)\to X(i)$ is injective, and the left $R$-modules $X(i)$ and ${\rm Coker}\varphi_i^X$ are Gorenstein flat. As an application of this result, we show that there is a hereditary abelian model structure on ${\sf Rep}(Q,R)$ whose cofibrant objects are precisely the Gorenstein flat representations, fibrant objects are precisely the cotorsion representations, and trivial objects are precisely the representations with values in the right orthogonal category of all projectively coresolved Gorenstein flat left $R$-modules.