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

2015/06/02 by Polstra, Thomas
#13A35 #13D40 #13F40 #14B05 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1506.01073

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.

Related

• Uniform bounds in excellent F_p-algebras and applications to semi-continuity
• Lower bounds on Hilbert--Kunz multiplicities and maximal F-signatures
• Continuity of Hilbert-Kunz multiplicity and F-signature
• When does the F-signature exist?
• Globalizing F-invariants