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LOCAL COHOMOLOGY UNDER SMALL PERTURBATIONS

Part of: Commutative algebra: Homological methods Theory of modules and ideals

Published online by Cambridge University Press:  20 January 2025

LUÍS DUARTE Open the ORCID record for LUÍS DUARTE [Opens in a new window]
LUÍS DUARTE*
Affiliation:
Department of Mathematics Stockholm University SE-106 91 Stockholm, Sweden
*
luis.duarte@math.su.se
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Abstract

Let $(R,\mathfrak {m})$ be a Noetherian local ring and I an ideal of R. We study how local cohomology modules with support in $\mathfrak {m}$ change for small perturbations J of I, that is, for ideals J such that $I\equiv J\bmod \mathfrak {m}^N$ for large N, under the hypothesis that $R/I$ and $R/J$ share the same Hilbert function. As one of our main results, we show that if $R/I$ is generalized Cohen–Macaulay, then the local cohomology modules of $R/J$ are isomorphic to the corresponding local cohomology modules of $R/I$, except possibly the top one. In particular, this answers a question raised by Quy and V. D. Trung. Our approach also allows us to prove that if $R/I$ is Buchsbaum, then so is $R/J$. Finally, under some additional assumptions, we show that if $R/I$ satisfies Serre’s property $(S_n)$, then so does $R/J$.

Keywords

perturbationHilbert functionlocal cohomologyinitial moduleBuchsbaum ring

MSC classification

Primary: 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 13D45: Local cohomology
Secondary: 13C05: Structure, classification theorems 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Information

Type
Article
Information
Nagoya Mathematical Journal , Volume 258 , June 2025 , pp. 183 - 200
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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