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

Abstract

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.

Global Frobenius Betti numbers and F-splitting ratio

“…The upper semi-continuity was first established in [Kun76] (Kunz claimed that this result was true for an equidimensional ring, but Shepherd-Barron noted in [SB79] that the locally equidimensional assumption is needed). The uniform convergence was established in [Pol18,Theorem 5.1]. An immediate consequence of these two facts is that the Hilbert-Kunz function is upper semi-continuous on the spectrum of a locally equidimensional ring [Smi16,Pol18].…”

-Signature Under Birational Morphisms

“…where λ R (•) denotes the length of the module •. We refer the reader to [HL02,Pol18,PT18,Tuc12] for additional properties of F -signature.…”

F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison

“…In this article, we will be primarily concerned with two important numerical invariants that measure the failure of flatness for the iterated Frobenius: the Hilbert-Kunz multiplicity [Mon83] and the F -signature [SVdB97,HL02]. Our aim is to revisit a number of core results about these invariants -existence [Tuc12], semicontinuity [Smi16,Pol15], positivity [HH94, AL03] -and provide vastly simplified proofs, which in turn yield new and important results. In particular, we confirm the suspicion of Watanabe and Yoshida allowing the F -signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals in a local ring [WY04, Question 1.10].…”

“…Aberbach and Leuschke [AL03] have shown that the positivity of the F -signature characterizes the notion of strong F -regularity introduced by Hochster and Huneke [HH94] in their celebrated study of tight closure [HH90]. Recently, it has been shown by the first author that the F -signature determines a lower semicontinuous R-valued function on ring spectra [Pol15]; an unpublished and independent proof was simultaneously found and shown to experts by the second author, and has been incorporated into this article.…”