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On the reducing projective dimension over local rings

Part of: Theory of modules and ideals Local rings and semilocal rings Commutative algebra: Homological methods

Published online by Cambridge University Press:  31 October 2023

Olgur Celikbas ,
Souvik Dey Open the ORCID record for Souvik Dey [Opens in a new window] ,
Toshinori Kobayashi  and
Hiroki Matsui
Olgur Celikbas
Affiliation:
School of Mathematical and Data Sciences, West Virginia University, Morgantown, WV, USA
Souvik Dey
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS, USA Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75 Praha, Czech Republic
Toshinori Kobayashi
Affiliation:
School of Science and Technology, Meiji University, Kawasaki-shi, Kanagawa, Japan
Hiroki Matsui*
Affiliation:
Department of Mathematical Sciences, Faculty of Science and Technology, Tokushima University, Tokushima, Japan
*
Corresponding author: Hiroki Matsui; Email: hmatsui@tokushima-u.ac.jp
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Abstract

In this paper, we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya–Celikbas and Araya–Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.

Keywords

G-regular ringsminimal multiplicityreducing projective and reducing Gorenstein dimension

MSC classification

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13C12: Torsion modules and ideals 13D05: Homological dimension 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 1 , January 2024 , pp. 104 - 118
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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Footnotes

Souvik Dey was partly supported by Charles University Research Center program No.UNCE/SCI/022 and a grant GA ČR 23-05148S from the Czech Science Foundation; Toshinori Kobayashi was partly supported by JSPS Grant-in-Aid for JSPS Fellows 18J20660; Hiroki Matsui was partly supported by JSPS Grant-in-Aid for Early-Career Scientists 22K13894.

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