Cotorsion pairs in categories of quiver representations

Abstract

We study the category $Rep(Q,\mathcal{M})$ 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 $(\Phi (\mathcal{A}),Rep(Q,\mathcal{B}\left)\right)$ and $(Rep(Q,\mathcal{A}),\Psi (\mathcal{B}\left)\right)$ in $Rep(Q,\mathcal{M})$. This is akin to a result by Gillespie, which asserts that a cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces cotorsion pairs $(\tilde{\mathcal{A}},dg\tilde{\mathcal{B}})$ and $(dg\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ in the category $Ch\left(\mathcal{M}\right)$ of chain complexes in $\mathcal{M}$. Special cases of our results recover descriptions of the projective and injective objects in $Rep(Q,\mathcal{M})$ proved by Enochs, Estrada, and García Rozas.