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.
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Communicated by B. Siebert.
This project was supported by Alexander von Humboldt Stiftung and National Natural Science Foundation of China (No. 10971206).
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Chen, XW. Unifying two results of Orlov on singularity categories. Abh. Math. Semin. Univ. Hambg. 80, 207–212 (2010). https://doi.org/10.1007/s12188-010-0044-x
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DOI: https://doi.org/10.1007/s12188-010-0044-x