Unifying two results of Orlov on singularity categories

Abstract

Let \(\mathbb{X}\) be a separated Noetherian scheme 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 triangulated 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 Orlov. We also prove a noncommutative version of this result.