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ANNIHILATORS AND DIMENSIONS OF THE SINGULARITY CATEGORY

Part of: Abelian categories Representation theory of rings and algebras Commutative algebra: Homological methods

Published online by Cambridge University Press:  06 January 2023

JIAN LIU Open the ORCID record for JIAN LIU [Opens in a new window]
JIAN LIU*
Affiliation:
School of Mathematics and Statistics Central China Normal University Wuhan 430079 China
*
liuj231@sjtu.edu.cn
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Abstract

Let R be a commutative Noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the annihilator of the singularity category of R coincides with the Jacobian ideal of R up to radical. We establish a relationship between the annihilator of the singularity category of R and the cohomological annihilator of R under some mild assumptions. Finally, we give an upper bound for the dimension of the singularity category of an equicharacteristic excellent local ring with isolated singularity. This extends a result of Dao and Takahashi to non-Cohen–Macaulay rings.

Keywords

annihilator of the singularity categorycohomological annihilatorgeneration timeJacobian idealKoszul object

MSC classification

Primary: 13D09: Derived categories
Secondary: 13D07: Homological functors on modules (Tor, Ext, etc.) 16G50: Cohen-Macaulay modules 18E35: Localization of categories
Type
Article
Information
Nagoya Mathematical Journal , Volume 250 , June 2023 , pp. 533 - 548
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Copyright
The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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