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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Uniform bounds in F-finite rings and lower semi-continuity of the F-signature
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by Thomas Polstra PDF
Trans. Amer. Math. Soc. 370 (2018), 3147-3169 Request permission


This paper establishes uniform bounds in characteristic $p$ rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the F-signature function. From this we establish that the F-signature function is lower semi-continuous. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, which was originally proven by Ilya Smirnov.
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Additional Information
  • Thomas Polstra
  • Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
  • Email:
  • Received by editor(s): June 2, 2015
  • Received by editor(s) in revised form: September 24, 2015, and July 21, 2016
  • Published electronically: December 19, 2017
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3147-3169
  • MSC (2010): Primary 13A35, 13D40, 13F40, 14B05
  • DOI:
  • MathSciNet review: 3766845