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Uniform bounds in F-finite rings and lower semi-continuity of the F-signature


Author: Thomas Polstra
Journal: Trans. Amer. Math. Soc. 370 (2018), 3147-3169
MSC (2010): Primary 13A35, 13D40, 13F40, 14B05
DOI: https://doi.org/10.1090/tran/7030
Published electronically: December 19, 2017
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Abstract: 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: tmpxv3@mail.missouri.edu

DOI: https://doi.org/10.1090/tran/7030
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
Article copyright: © Copyright 2017 American Mathematical Society

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