Cotorsion pairs in categories of quiver representations

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.