Unifying two results of D. Orlov

TL;DR

This paper unifies two of D. Orlov's results by establishing an equivalence between the singularity category of an open subscheme and a quotient of the singularity category of the ambient scheme, extending to noncommutative cases.

Contribution

It provides a unified framework for understanding singularity categories of open subschemes and generalizes the result to noncommutative settings.

Findings

Singularity category of $U$ is equivalent to a Verdier quotient of that of $X$.

The result applies to schemes with enough locally free sheaves of finite rank.

A noncommutative analogue of the main theorem is also established.

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.