On resurgence via asymptotic resurgence

The resurgence and asymptotic resurgence of an ideal in a polynomial ring are two statistics which measure the relationship between its regular and symbolic powers. We address two aspects of resurgence which can be studied via asymptotic resurgence. First, we show that if an ideal has Noetherian sym...

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Bibliographic Details
Published in:Journal of algebra 2021-12, Vol.587, p.64-84
Main Authors: DiPasquale, Michael, Drabkin, Ben
Format: Article
Language:English
Subjects:
Asymptotic resurgence
Containment problem
Resurgence
Symbolic powers
Symbolic Rees algebra
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Summary:The resurgence and asymptotic resurgence of an ideal in a polynomial ring are two statistics which measure the relationship between its regular and symbolic powers. We address two aspects of resurgence which can be studied via asymptotic resurgence. First, we show that if an ideal has Noetherian symbolic Rees algebra then its resurgence is rational. Second, we derive two bounds on asymptotic resurgence given a single known containment between a symbolic and regular power. From these bounds we recover and extend criteria for the resurgence of an ideal to be strictly less than its big height recently derived by Grifo, Huneke, and Mukundan. We achieve the reduction to asymptotic resurgence by showing that if the asymptotic resurgence and resurgence are different, then resurgence is a maximum instead of a supremum.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2021.07.021