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arXiv · 1506.01073

Uniform Bounds in F-Finite Rings and Lower Semi-Continuity of the F-Signature

Thomas Polstra
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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. 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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Thomas Polstra. 2015-06-02. Uniform Bounds in F-Finite Rings and Lower Semi-Continuity of the F-Signature. https://arxiv.org/abs/1506.01073

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