Special precovered categories of Gorenstein categories

Abstract

Let A be an abelian category and P(A ) the subcategory of A consisting of projective objects. Let C be a full, additive and self-orthogonal subcategory of A with P(A ) a generator, and let G(C ) be the Gorenstein subcategory of A . Then the right 1-orthogonal category G(C ) ⊥ As a generalization of finitely generated projective modules, Auslander and Bridger introduced in [3] the notion of finitely generated modules of Gorenstein dimension zero over commutative Noetherian rings.Then Enochs and Jenda generalized it in [7] to arbitrary modules over a general ring and introduced the notion of Gorenstein projective modules and its dual (that is, the notion of Gorenstein injective modules). Let A be an abelian category and C an additive and full subcategory of A . Recently Sather-Wagstaff, Sharif and White introduced in [14] the notion of the Gorenstein subcategory G(C ) of A , which is a common generalization of the notions of modules of Gorenstein dimension zero [3], Gorenstein projective modules, Gorenstein injective modules [7], V -Gorenstein projective modules and V -Gorenstein injective modules [9], and so on. Let R be an associative ring with identity, and let Mod R be the category of left R-modules and G(P(Mod R) the subcategory of Mod R consisting of Gorenstein projective modules. Let PC(G(P(Mod R))and SPC(G(P(Mod R)) be the subcategories of Mod R consisting of modules admitting a G(P(Mod R))precover and admitting a special G(P(Mod R))-precover respectively. The following question in relative homological algebra remains still open: does PC(G(P(Mod R)) = Mod R always hold true? Several authors have gave some partially positive answers to this question, see [2,4,5,16]. Note that in these references, PC(G(P(Mod R)) = SPC(G(P(Mod R)), see Example 4.8 below for details. In particular, any module in Mod R with finite Gorenstein projective dimension admits a G(P(Mod R))-precover which is * 2010 Mathematics Subject Classification: 18G25, 18E10. †

Subprojectivity in abelian categories

“…In the last one we show that every object in GP ⊥ has a special GP-precover. This is in fact the recent result [24,Proposition 4.1] established in a different way.…”

“…The question of whether or not any object has a special GP-precover has been a subject of many papers. Here, as a consequence of Corollary 2.29 and Theorem 4.1 (since it is known that the class GP is closed under kernels of epimorphisms) we immediately get a partial answer which has been recently known following different methods (see [24,Proposition 4.1]). Corollary 4.3.…”