Gorenstein Projective Representations Relative to Cotorsion Pairs

设 Q 是一个有限无圈箭图, Q 是一个环, R e p ( Q , R ) 是 Q 的左 R -模表示范畴。本文描述了 R e p ( Q , R ) 中相对于余挠对的Gorenstein投射表示。具体地，我们证明了 X ∈ R e p ( Q , R ) 是相对于 ( A , B ) 的Gorenstein投射表示当且仅当对任意的 i ∈ Q 0 $ , $ φ i X : ⨁ a ∈ Q 1 ∗ → i X ( s ( a ) ) → X ( i ) 是单同态,并且 X ( i ) , C o k e r φ i X 是相对于 ( A , B ) 的Gorenstein投射模.Let Q be a finite acyclic quiver, R a ring, and R e p ( Q , R ) the category of left R -module representations of Q . This paper characterizes Gorenstein projective representations in R e p ( Q , R ) relative to cotorsion pairs. Specifically, we prove that a representation X ∈ Rep ( Q , R ) is Gorenstein projective relative to ( A , B ) if and only if} for every vertex i ∈ Q 0 , , the canonical morphism φ i X : ⨁ a ∈ Q 1 ∗ → i X ( s ( a ) ) → X ( i ) is a monomorphism, and both X ( i ) and C o k e r φ i X are Gorenstein projective modules relative to ( A , B ) .