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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Castelnuovo-Mumford regularity and the discreteness of $F$-jumping coefficients in graded rings
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by Mordechai Katzman and Wenliang Zhang PDF
Trans. Amer. Math. Soc. 366 (2014), 3519-3533 Request permission

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
  • MR Author ID: 805625
  • Email: wzhang15@unl.edu
  • 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.
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3519-3533
  • MSC (2010): Primary 13A35
  • DOI: https://doi.org/10.1090/S0002-9947-2014-05918-7
  • MathSciNet review: 3192605