Relative global Gorenstein dimensions

Abstract

Let $\mathcal{A}$ be an abelian category. In this paper, we investigate the global $(\mathcal{X},\mathcal{Y})$-Gorenstein projective dimension $gl.GPD(\mathcal{X},\mathcal{Y})(\mathcal{A})$, 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 $gl.GID(\mathcal{Z},\mathcal{W})(\mathcal{A})$ and $gl.GPD(\mathcal{X},\mathcal{Y})(\mathcal{A})$ are the same.