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Cohomological dimensions of specialization-closed subsets and subcategories of modules

Authors: Hiroki Matsui, Tran Tuan Nam, Ryo Takahashi, Nguyen Minh Tri and Do Ngoc Yen
Journal: Proc. Amer. Math. Soc. 149 (2021), 481-496
MSC (2020): Primary 13C60, 13D09, 13D45
Published electronically: December 18, 2020
MathSciNet review: 4198059
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Abstract: Let $R$ be a commutative noetherian ring. In this paper, we study specialization-closed subsets of $\mathrm {Spec} R$. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer $n$ we introduce the notion of $n$-wide subcategories of $R$-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most $n$.

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Additional Information

Hiroki Matsui
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
MR Author ID: 1103089
ORCID: 0000-0001-9545-8337

Tran Tuan Nam
Affiliation: Department of Mathematics-Informatics, Ho Chi Minh University of Pedagogy, 280 An Duong Vuong, District 5, Ho Chi Minh city, Vietnam
MR Author ID: 674002

Ryo Takahashi
Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
MR Author ID: 674867

Nguyen Minh Tri
Affiliation: Department of Natural Science Education, Dong Nai University, Dong Nai, Vietnam
MR Author ID: 1058394
ORCID: 0000-0002-1644-5069

Do Ngoc Yen
Affiliation: Faculty of Mathematics & Computer Science, University of Science, VNU-HCM, Ho Chi Minh City, Vietnam; and Faculty of Fundamental Science 2, Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam
MR Author ID: 1127982

Keywords: Big Cohen–Macaulay module, cohomological dimension, derived category, local cohomology, localizing subcategory, module category, Serre subcategory, support, specialization-closed subset, torsion functor, transform functor, wide subcategory
Received by editor(s): December 13, 2019
Received by editor(s) in revised form: February 29, 2020
Published electronically: December 18, 2020
Additional Notes: The first author was partly supported by JSPS Grant-in-Aid for JSPS Fellows 19J00158. The second, fourth, and fifth authors were funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.04-2018.304. The third author was partly supported by JSPS Grants-in-Aid for Scientific Research 16K05098 and JSPS Fund for the Promotion of Joint International Research 16KK0099.
Communicated by: Claudia Polini
Article copyright: © Copyright 2020 American Mathematical Society