Cotorsion pairs in categories of quiver representations

TL;DR

This paper explores how cotorsion pairs in an abelian category induce related cotorsion pairs in categories of quiver representations, extending known results and describing projective and injective objects.

Contribution

It establishes explicit methods to induce cotorsion pairs in quiver representation categories from those in the base category, generalizing Gillespie's results.

Findings

Induces cotorsion pairs in quiver representations from base category pairs

Provides explicit descriptions of induced cotorsion pairs

Recovers known results on projective and injective objects

Abstract

We study the category $\mathrm{Rep}(Q,\mathcal{M})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{M}$. Under certain assumptions, we show that every cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces two (explicitly described) cotorsion pairs $(\Phi(\mathcal{A}),\mathrm{Rep}(Q,\mathcal{B}))$ and $(\mathrm{Rep}(Q,\mathcal{A}),\Psi(\mathcal{B}))$ in $\mathrm{Rep}(Q,\mathcal{M})$. This is akin to a result by Gillespie, which asserts that a cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces cotorsion pairs $(\widetilde{\mathcal{A}}, \mathrm{dg}\,\widetilde{\mathcal{B}})$ and $(\mathrm{dg}\,\widetilde{\mathcal{A}}, \widetilde{\mathcal{B}})$ in the category $\mathrm{Ch}(\mathcal{M})$ of chain complexes in $\mathcal{M}$. Special cases of our results recover descriptions of the projective and injective objects in $\mathrm{Rep}(Q,\mathcal{M})$ proved by Enochs, Estrada, and Garcia Rozas.