2021/02/11 by Di, Zhenxing, Li, Liping, Liang, Li +1
#FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2102.05826
This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs (Φ(\sf X),Φ(\sf X)⊥) and (⊥Ψ(\sf Y),Ψ(\sf Y)) in Rep(Q,\sfA) -- the category of all \sfA-valued representations of a quiver Q -- are complete whenever (\sf X,\sf Y) is a complete cotorsion pair in an abelian category \sfA satisfying some mild conditions. Recently, Odabaşı gave an affirmative answer if the quiver Q is rooted and the cotorsion pair (\sf X,\sf Y) is further hereditary. In this paper, we improve Odabaşı's work by removing the hereditary assumption on the cotorsion pair. As an application, we show under certain mild conditions that if a subcategory \sf L, which is not necessarily closed under direct summands, of \sf A is special precovering (resp., preenveloping), then Φ(\sf L) (resp., Ψ(\sf L)) is special precovering (resp., preenveloping) in Rep(Q,\sfA).