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TILTING COMPLEXES AND CODIMENSION FUNCTIONS OVER COMMUTATIVE NOETHERIAN RINGS

Part of: Local rings and semilocal rings Commutative algebra: Homological methods

Published online by Cambridge University Press:  15 March 2024

MICHAL HRBEK Open the ORCID record for MICHAL HRBEK [Opens in a new window] ,
TSUTOMU NAKAMURA Open the ORCID record for TSUTOMU NAKAMURA [Opens in a new window]  and
JAN ŠŤOVÍČEK Open the ORCID record for JAN ŠŤOVÍČEK [Opens in a new window]
MICHAL HRBEK*
Affiliation:
Institute of Mathematics Czech Academy of Sciences Žitná 25 115 67 Prague Czech Republic
TSUTOMU NAKAMURA
Affiliation:
Department of Mathematics, Faculty of Education Mie University 1577 Kurimamachiya-cho Tsu, Mie 514-8507 Japan nakamura@edu.mie-u.ac.jp Osaka Central Advanced Mathematical Institute Osaka Metropolitan University 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585 Japan t.nakamura.math@gmail.com
JAN ŠŤOVÍČEK
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics Charles University Sokolovská 83 186 75, Praha Czech Republic stovicek@karlin.mff.cuni.cz
*
hrbek@math.cas.cz
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Abstract

In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.

Keywords

derived categorysilting objectcosilting objecttilting complexcommutative noetherian ring

MSC classification

Primary: 13D09: Derived categories
Secondary: 13D45: Local cohomology 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Type
Article
Information
Nagoya Mathematical Journal , Volume 255 , September 2024 , pp. 618 - 693
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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Footnotes

Dedicated to Lidia Angeleri Hügel on the occasion of her 60th birthday

M. Hrbek was supported by the GAČR project 20-13778S and RVO: 67985840. T. Nakamura was supported by PRIN-2017 “Categories, Algebras: Ring-Theoretical and Homological Approaches (CARTHA),” Grant-in-Aid for JSPS Fellows JP20J01865, and Grant-in-Aid for Early-Career Scientists JP23K12954. J. Šťovíček was supported by the GAČR project 20-13778S.

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