Cotorsion pairs in categories of quiver representations

摘要

We study the category Rep(Q,M) of representations of a quiver Q with values in an abelian categoryM. Under certain assumptions, we show that every cotorsion pair (A,B) inMinduces two (explicitly described) cotorsion pairs (Φ(A), Rep(Q,B)) and (Rep(Q,A),Ψ(B)) in Rep(Q,M).This is akin to a result byGillespie,which asserts that a cotorsion pair (A,B) inMinduces cotorsion pairs (?, dgB~) and (dg?,B~) in the category Ch(M) of chain complexes inM. Special cases of our results recover descriptions of the projective and injective objects in Rep(Q,M) proved by Enochs, Estrada, and García Rozas.