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. *

Abelian model structures on categories of quiver representations

“…The following is the main result of [26] and addresses the question of when the cotorsion pairs found in 2.7 and 2.8 are complete. Fact 2.9.…”

“…Fact 2.9. [26,Thm. 4.1.3] Let M be an abelian category as in setup 2.4 1 and let (A, B) be a complete cotorsion pair in M. Then the following hold:…”

Gorenstein flat representations of left rooted quivers

“…3.8] by Gillespie on cotorsion pairs in the categories of chain complexes. Odabaşı further showed in [19] that the two induced cotorsion pairs are complete as well. These wonderful results help us to obtain the following result, which asserts that a hereditary Hovey triple (C, W, F) on A can induce a hereditary Hovey triple on Rep(Q, A) with trivial objects are precisely the representations with values in W (here by "hereditary Hovey triple" we mean that the two associated cotorsion pairs in Hovey's correspondence are hereditary); see Section 1 for unexplained notation.…”

Completeness of the induced cotorsion pairs in functor categories

“…The following is a general form of the so-called Salce's Lemma (see [11]), which can be found in [10]. 3.2 Lemma.…”