Unifying two results of D. Orlov

Abstract

Let $\mathbb{X}$ be a noetherian separated scheme $\mathbb{X}$ of finite Krull dimension which has enough locally free sheaves of finite rank and let $U\subseteq \mathbb{X}$ be an open subscheme. We prove that the singularity category of $U$ is triangle equivalent to the Verdier quotient category of the singularity category of $\mathbb{X}$ with respect to the thick triangulated subcategory generated by sheaves supported in the complement of $U$. The result unifies two results of D. Orlov. We also prove a noncommutative version of this result.