  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.
