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Castelnuovo-Mumford regularity and the discreteness of $ F$-jumping coefficients in graded rings


Authors: Mordechai Katzman and Wenliang Zhang
Journal: Trans. Amer. Math. Soc. 366 (2014), 3519-3533
MSC (2010): Primary 13A35
DOI: https://doi.org/10.1090/S0002-9947-2014-05918-7
Published electronically: March 4, 2014
MathSciNet review: 3192605
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Abstract: In this paper we show that the sets of $ F$-jumping coefficients of ideals form discrete sets in certain graded $ F$-finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of certain ideals. We further show that these linear bounds exist for one-dimensional rings and for ideals of (most) two-dimensional domains. We conclude by applying our technique to prove that all sets of $ F$-jumping coefficients of all ideals in the determinantal ring given as the quotient by $ 2\times 2$ minors in a $ 2\times 3$ matrix of indeterminates form discrete sets.


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Additional Information

Mordechai Katzman
Affiliation: Department of Pure Mathematics, Hicks Building, University of Sheffield, Sheffield S3 7RH, United Kingdom
Email: M.Katzman@sheffield.ac.uk

Wenliang Zhang
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
Email: wzhang15@unl.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05918-7
Received by editor(s): October 17, 2011
Received by editor(s) in revised form: July 10, 2012
Published electronically: March 4, 2014
Additional Notes: The results in this paper were obtained while both authors enjoyed the hospitality of the School of Mathematics at the University of Minnesota. The first author also wishes to acknowledge support through Royal Society grant TG102669. The second author was supported in part by NSF Grant DMS #1068946.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.