Relative singularity categories II

Abstract

Let $\mathscr{A}$ be an abelian category with enough projective objects and $\mathscr{C}$ an additive and full subcategory of $\mathscr{A}$, and let $\mathscr{G}(\mathscr{C})$ be the Gorenstein category of $\mathscr{C}$. We study the properties of the $\mathscr{C}$-derived category $D_\mathscr{C}^b(\mathscr{A})$, $\mathscr{C}$-singularity category $D_{\mathscr{C}-sg}(\mathscr{A})$ and $\mathscr{G}(\mathscr{C})$-defect category $D_{\mathscr{G(C)}-defect}(\mathscr{A})$ of $\mathscr{A}$. Let $\mathscr{C}$ be admissible in $\mathscr{A}$. We show that $D_{\mathscr{G(C)}-defect}(\mathscr{A})\simeq D_{\mathscr{C}-sg}(\mathscr{A})$ if and only if $\mathscr{C}=\mathscr{G(C)}$; and $D_{\mathscr{G(C)}-defect}(\mathscr{A})=0$ if and only if the stable category $\underline{\mathscr{G}(\mathscr{C})}$ of $\mathscr{G}(\mathscr{C})$ is triangle-equivalent to $D_{\mathscr{C}-sg}(\mathscr{A})$, and if and only if every object in $\mathscr{A}$ has finite $\mathscr{C}$-proper $\mathscr{G}(\mathscr{C})$-dimension. Then we apply these results to module categories. We prove that under some condition, the Gorenstein derived equivalence of artin algebras induces the Gorenstein singularity equivalence. Finally, for an artin algebra $A$, we establish the stability of Gorenstein defect categories of $A$.