Relative global Gorenstein dimensions

TL;DR

This paper explores the properties of global Gorenstein dimensions in abelian categories, providing homological characterizations and conditions for equivalence between different Gorenstein dimensions.

Contribution

It introduces homological conditions that characterize global Gorenstein projective and injective dimensions associated with GP- and GI-admissible pairs in abelian categories.

Findings

Homological conditions for global Gorenstein projective dimension

Criteria for equality of Gorenstein injective and projective dimensions

Characterizations applicable to abelian categories

Abstract

Let $\mathcal{A}$ be an abelian category. In this paper, we investigate the global $(\mathcal{X} , \mathcal{Y})$-Gorenstein projective dimension $\mathrm{gl.GPD}(\mathcal{X} ,\mathcal{Y})(\mathcal{A})$, associated to a GP-admissible pair $(\mathcal{X} , \mathcal{Y} )$. We give homological conditions over $(\mathcal{X} , \mathcal{Y})$ that characterize it. Moreover, given a GI-admisible pair $(\mathcal{Z} , \mathcal{W} )$, we study conditions under which $\mathrm{gl.GID}(\mathcal{Z},\mathcal{W})(\mathcal{A})$ and $\mathrm{gl.GPD}(\mathcal{X},\mathcal{Y})(\mathcal{A})$ are the same.