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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. More precisely, we first give some characterizations of ξ-Gprojective dimension by using derived functors on C. Second, let P(ξ) (resp. I(ξ)) be a generating (resp. cogenerating) subcategory of C. We show that the following equality holds under some assumptions:where ξ-GpdM (resp. ξ-GidM ) denotes ξ-Gprojective (resp. ξ-Ginjective) dimension of M . As an application, our main results generalize their work by Bennis-Mahdou and Ren-Liu. Moreover, our proof is not far from the usual module or triangulated case.

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In this article, we prove that if (A, B, C) is a recollement of extriangulated categories, then n-cotorsion pairs in A and C can induce n-cotorsion pairs in B. Conversely, this holds true under natural assumptions. Besides, we give mild conditions on a pseudo cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding additive quotients to form a recollement of semi-abelian categories.

In this article, we prove that if (A, B, C) is a recollement of extriangulated categories, then torsion pairs in A and C can induce torsion pairs in B, and the converse holds under natural assumptions. Besides, we give mild conditions on a cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding abelian quotients to form a recollement of abelian categories.