E-三角范畴中的(n, m)-强ξ-Gorenstein投射对象

(n, m)-Strongly ξ-Gorenstein Projective Objects in Extriangulated Categories

设C是一个E-三角范畴,ξ是C中的一个E-三角真类。在C中引入(n, m)-强ξ-Gorenstein投射对象的概念,研究了C中的对象与其合冲的这种ξ-Gorenstein投射性质之间的联系。作为应用,证明了ξ中对象X的ξ-Gorenstein投射维数小于等于m当且仅当存在C中的ξ-Gorenstein投射对象G,使得是(1, m)-强ξ-Gorenstein投射的。

Let C be an extriangulated category and ξ a proper class of E-triangles of C. The notion of (n, m)- strongly ξ-Gorenstein projective object in C is introduced and the relation of such ξ-Gorenstein projectivity of an object in C with that of its syzygies is investigated. As a consequence, it is shown that an object X of C has ξ-Gorenstein projective dimension at most m if and only if is (1, m)-strongly ξ-Gorenstein projective for some ξ-Gorenstein projective object of C.