Proper classes and Gorensteinness in extriangulated categories

Abstract

Let (C, E, s) be an extriangulated category with a proper class ξ of E-triangles, and W an additive full subcategory of C. We provide a method for constructing a proper W(ξ)-resolution (respectively, coproper W(ξ)-coresolution) of one term in an E-triangle in ξ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category GW(ξ) in extriangulated categories. These results generalise their work by Huang and Yang-Wang, but the proof is not too far from their case. Finally, we give some applications about our main results.

Relative homological dimensions in recollements of triangulated categories

“…Similar to the way of defining E-projective and E-injective dimensions, for an object A ∈ C, the E-Gprojective dimension E-GpdA and E-Ginjective dimension E-GidA are defined inductively in [5].…”

“…In fact, with the modifications of the usual proof, one obtains the isomorphism ξxt n P(ξ) (A, B) ∼ = ξxt n I(E) (A, B), which is denoted by ξxt n ξ (A, B). It is easy to see that the following lemma holds by [5,Lemma 4.14].…”

“…The notion of extriangulated categories was introduced by Nakaoka and Palu in [6] as a simultaneous generalization of exact categories and triangulated categories. Exact categories and extension closed subcategories of an extriangulated category are extriangulated categories, while there exist some other examples of extriangulated categories which are neither exact nor triangulated, see [6,8,5]. Hence many results hold on exact categories and triangulated categories can be unified in the same framework.…”

“…Let (C, E, s) be an extriangulated category. The authors [5] studied a relative homological algebra in C which parallels the relative homological algebra in a triangulated category. By specifying a class of E-triangles, which is called a proper class ξ of E-triangles, we introduced ξ-Gprojective dimensions and ξ-Ginjective dimensions, and discussed their properties.…”

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Let (C, E, s) be an extriangulated category with a proper class ξ of E-triangles. The authors introduced and studied ξ-Gprojective and ξ-Ginjective in [5]. In this paper, we discuss Gorenstein homological dimensions for extriangulated categories.

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