Completeness of the induced cotorsion pairs in categories of quiver representations

Abstract

Given a complete hereditary cotorsion pair (A,B) in an abelian category C satisfying certain conditions, we study the completeness of the induced cotorsion pairs (Φ(A),Φ(A)⊥) and (Ψ⊥(B),Ψ(B)) in the category Rep(Q,C) of C-valued representations of a given quiver Q. We show that if Q is left rooted, then the cotorsion pair (Φ(A),Φ(A)⊥) is complete, and if Q is right rooted, then the cotorsion pair (Ψ⊥(B),Ψ(B)) is complete. Besides, we work on the infinite line quiver A∞∞, which is neither left rooted nor right rooted. We prove that these cotorsion pairs in Rep(A∞∞,R) are complete, as well.