Relative Singularity Categories

Abstract

We study the properties of the relative derived category $D_{\mathcal{C}}^b$($\mathcal{A}$) of an abelian category $\mathcal{A}$ relative to a full and additive subcategory $\mathcal{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$ for a finite-dimensional algebra $A$ over a field and $\mathcal{C}$ is a contravariantly finite subcategory of $A$-$\mod$ which is admissible and closed under direct summands, the $\mathcal{C}$-singularity category $D_{\mathcal{C}\text{s}g}$($\mathcal{A}$)=$D_{\mathcal{C}}^b$($\mathcal{A}$)/$K^b(\mathcal{C})$ is studied. We give a sufficient condition when this category is triangulated equivalent to the stable category of the Gorenstein category $\mathcal{G}(\mathcal{C})$ of $\mathcal{C}$.

17 pages, to appear in Journal of Pure and Applied Algebra