$\V\W$-Gorenstein categories

Abstract

Let A be an abelian category, and V , W two additive full subcategories of A . We introduce and study the VW -Gorenstein subcategory of A , which unifies many known notions, such as the Gorenstein category and the category consisting of GC -projective (injective) modules, although they were defined in a different way. We also prove that the Bass class with respect to a semidualizing module is one kind of VW -Gorenstein category. The connections between VW -Gorenstein categories and Gorenstein categories are discussed. Some applications are given.

$VW$-Gorenstein complexes

“…by [29,Corollary 3.8]. Thus Ext 1 C (V, K 0 ) = 0 and Ext 1 C (X, W ) = 0 for any V ∈ V, W ∈ W by Corollary 3.5.…”

“…where all G n i are VW -Gorenstein by Theorem 3.8. Thus X n is a VW -Gorenstein module by [29,Theorem 4.2]. Therefore, X is VW -Gorenstein by Theorem 3.8.…”

“…Then there is an exact sequence 0 Ext 1 (Z n , W ) = 0 for any W ∈ W . Thus [29,Proposition 3.10], and so X ∈ G( V W) by Theorem 3.8.…”

“…where V n ∈ V , and Y n ∈ G(VW) by [29,Corollary 4.6]. Thus we have the following short exact sequence:…”

“…[3-5, 9, 12, 14, 15, 17, 20, 21, 23, 25]. Recently, Zhao and Sun [29] introduced the notion of VW -Gorenstein modules, where V and W are two classes of modules. This notion unifies the following notions: Gorenstein projective and Gorenstein injective modules [8], G C -projective and G C -injective modules [20], W -Gorenstein modules [12,21], and so on.…”