相对于余挠对的Gorenstein投射表示

Gorenstein Projective Representations Relative to Cotorsion Pairs

摘要: 设 Q 是一个有限无圈箭图,R是一个环,Rep(Q,R)是 Q 的左 R- 模表示范畴。本文描述了 Rep(Q,R) 中相对于余挠对的 Gorenstein 投射表示。具体地，我们证明了X∈Rep(Q,R) 是相对于(A,B)的 Gorenstein 投射表示当且仅当对任意的i∈𝑄₀,$${\phi }_i^X:\underset{a\in Q_1^{\ast \to i}}{⨁}X(s(a))\to X(i)$$是单同态,并且X(i),Cokerφ_i^X是相对于 (A,B)的Gorenstein投射模.

Abstract: Let Q be a finite acyclic quiver, R a ring, and Rep(Q,R) the category of left R-module representations of Q. This paper characterizes Gorenstein projective representations in Rep(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∈Q₀, the canonical morphism $${\phi }_i^X:\underset{a\in Q_1^{\ast \to i}}{⨁}X(s(a))\to X(i)$$ is a monomorphism, and both X(i) and Coker φ_i ^X are Gorenstein projective modules relative to (A,B).