## 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

## 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.## References

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## Additional Information

**Thomas Polstra**- Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
- Email: tmpxv3@mail.missouri.edu
- 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: https://doi.org/10.1090/tran/7030
- MathSciNet review: 3766845