Completeness of the induced cotorsion pairs in categories of quiver representations

Original · EN

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.