The derived category with respect to $\boldsymbol{\mathcal{G}(\mathcal{X})}$

Abstract

Let $\\mathcal{A}$ be an abelian category with enough projective objects and $(\\mathcal{X},~\\mathcal{Y})$ be a complete and hereditary cotorsion pair in $\\mathcal{A}$. We introduce the Gorenstein $\\mathcal{X}$-derived category, denoted by $\\mathbf{D}_{\\mathcal{G}(\\mathcal{X})}{(\\mathcal{A})}$, and investigate $\\mathbf{D}_{\\mathcal{G}(\\mathcal{X})}{(\\mathcal{A})}$ from different perspectives. When $(\\mathcal{X},~\\mathcal{Y})$ is served as a special cotorsion pair, we obtain some known derived categories, such as Gorenstein derived categories, Gorenstein flat derived categories and so on. The bounded Gorenstein $\\mathcal{X}$-derived category $\\mathbf{D}^{b}{_{\\mathcal{G}(\\mathcal{X})}{(\\mathcal{A})}}$ and bounded derived category $\\mathbf{D}^{b}{(\\mathcal{A})}$ are described via the homology category $\\mathbf{K}^{-,gxb}{(\\mathcal{G}(\\mathcal{X}))}$ related to $\\mathcal{G}(\\mathcal{X})$ and some triangle equivalences are given.