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CONSTANCY OF THE HILBERT–SAMUEL FUNCTION

Part of: Birational geometry Surfaces and higher-dimensional varieties Local theory Arithmetic rings and other special rings

Published online by Cambridge University Press:  24 May 2024

VINCENT COSSART Open the ORCID record for VINCENT COSSART [Opens in a new window] ,
OLIVIER PILTANT Open the ORCID record for OLIVIER PILTANT [Opens in a new window]  and
BERND SCHOBER Open the ORCID record for BERND SCHOBER [Opens in a new window]
VINCENT COSSART
Affiliation:
Laboratoire de Mathématiques de Versailles LMV UMR 8100 Université de Paris-Saclay 45, avenue des États-Unis 78035 VERSAILLES Cedex France vincent.cossart@uvsq.fr
OLIVIER PILTANT
Affiliation:
Laboratoire de Mathématiques de Versailles LMV UMR 8100 Université de Paris-Saclay 45, avenue des États-Unis 78035 VERSAILLES Cedex France olivier.piltant@uvsq.fr
BERND SCHOBER*
Affiliation:
Institut für Mathematik Carl von Ossietzky, Universität Oldenburg 26111 Oldenburg (Oldb) Germany Current affiliation: None. (Hamburg, Germany)
*
schober.math@gmail.com
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Abstract

We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.

Keywords

excellent schemesHilbert–Samuel functionsingularitiesresolution of singularities

MSC classification

Primary: 14B05: Singularities 14E15: Global theory and resolution of singularities 14J17: Singularities 13F40: Excellent rings
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 15
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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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