Relative Singularity Categories

TL;DR

This paper investigates the properties of relative derived and singularity categories in abelian categories, establishing conditions under which these categories are triangulated equivalent to stable categories of Gorenstein categories, thus advancing the understanding of relative homological algebra.

Contribution

It introduces new conditions for the triangulated equivalence between relative singularity categories and stable Gorenstein categories in the context of finite-dimensional algebras.

Findings

Established a sufficient condition for triangulated equivalence.

Characterized the structure of the relative singularity category.

Extended the understanding of Gorenstein categories in relative settings.

Abstract

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