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ANALYTIC SPREAD OF FILTRATIONS ON TWO-DIMENSIONAL NORMAL LOCAL RINGS

Part of: General commutative ring theory Local theory

Published online by Cambridge University Press:  23 November 2022

STEVEN DALE CUTKOSKY Open the ORCID record for STEVEN DALE CUTKOSKY [Opens in a new window]
STEVEN DALE CUTKOSKY*
Affiliation:
Department of Mathematics University of Missouri Columbia, Missouri 65211 USA [email protected]
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Abstract

In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.

MSC classification

Primary: 14B05: Singularities
Secondary: 13A18: Valuations and their generalizations 14B25: Local structure of morphisms
Type
Article
Information
Nagoya Mathematical Journal , Volume 249 , March 2023 , pp. 239 - 268
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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Footnotes

Partially supported by National Science Foundation Grant No. 2054394.

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