Relative singularity categories, Gorenstein objects and silting theory

Abstract

We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let $\omega$ be a semi-selforthogonal (or presilting) subcategory of a triangulated category $\mathcal{T}$. We introduce the notion of $\omega$-Gorenstein objects, which is far extended version of Gorenstein projective modules and Gorenstein injective modules in triangulated categories. We prove that the stable category $\underline{{\mathcal{G}}_{\omega }}$, where ${\mathcal{G}}_{\omega }$ is the subcategory of all $\omega$-Gorenstein objects, is a triangulated category and it is, under some conditions, triangle equivalent to the relative singularity category of $\mathcal{T}$ with respect to $\omega$.