Completeness of the induced cotorsion pairs in categories of quiver representations

Abstract

Given a complete hereditary cotorsion pair $(\mathcal{A},\mathcal{B})$ in an abelian category $\mathcal{C}$ satisfying certain conditions, we study the completeness of the induced cotorsion pairs $(\Phi (\mathcal{A}),\Phi (\mathcal{A}{)}^{\perp })$ and ${(}^{\perp }\Psi (\mathcal{B}),\Psi (\mathcal{B}))$ in the category $\mbox{Rep}(Q, \mathcal{C})$ of $\mathcal{C}$-valued representations of a given quiver $Q$. We show that if $Q$ is left rooted, then the cotorsion pair $(\Phi (\mathcal{A}),\Phi (\mathcal{A}{)}^{\perp })$ is complete, and if $Q$ is right rooted, then the cotorsion pair ${(}^{\perp }\Psi (\mathcal{B}),\Psi (\mathcal{B}))$ is complete. Besides, we work on the infinite line quiver $A_{\infty }^{\infty }$, which is neither left rooted nor right rooted. We prove that these cotorsion pairs in $\mbox{Rep}(A_{\infty}^{\infty}, R)$ are complete, as well.