Upper semi-continuity of the Hilbert–Kunz multiplicity

Smirnov, Ilya

doi:10.1112/s0010437x15007800

Cited by 17 publications

(21 citation statements)

References 15 publications

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“…Uniform convergence and semi-continuity results. In this subsection we recall some results proved by Smirnov in [Smi16], the second author in [Pol], and the second author and Tucker in [PT16] that will be of use in later sections. The following theorem is the analogue of Theorem 2.34 for the sequence of normalized Frobenius splitting number functions.…”

Section: 2mentioning

confidence: 99%

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Globalizing F-invariants

Stefani

Polstra

Yao

2019

Advances in Mathematics

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In this paper we define and study the global Hilbert-Kunz multiplicity and the global F-signature of prime characteristic rings which are not necessarily local. Our techniques are made meaningful by extending many known theorems about Hilbert-Kunz multiplicity and F-signature to the non-local case.1 Now let R be an F-finite domain, not necessarily local. With the above observation, we define the global Hilbert-Kunz multiplicity of R, still denoted e HK (R), as e HK (R) = lim e→∞ µ(F e * R)/ rank(F e * R), provided the limit exists. Our first main result is the existence of the corresponding limit for any F-finite ring. In addition, we relate e HK (R) with the Hilbert-Kunz multiplicities e HK (R P ) of the localizations at primes P ∈ Spec(R), showing that such an invariant, even though it is defined globally, captures the local properties of the ring. Finally, as for the Hilbert-Kunz multiplicity of a local ring, we show that small values of e HK (R) imply that R has mild singularities. We summarize all these results in the following theorem. We point out that our results hold in a more general setup than the one in which we state them here, as we will show in Section 3.The inequality in the middle follows from the fact that the minimal number of generators of the T -module F e * M ′ can only decrease after localization at the prime Q. 6 7

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Section: 2mentioning

confidence: 99%

“…Thus Smirnov's theorem on the upper semi-continuity of Hilbert-Kunz multiplicity holds for finitely generated modules. See[Smi16, Main Theorem].Corollary 2.35. Let R be a locally equidimensional ring which is either F-finite or essentially of finite type over an excellent local ring and let M be a finitely generated R-module.…”

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confidence: 99%

Globalizing F-invariants

Stefani

Polstra

Yao

2019

Advances in Mathematics

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“…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].…”

Section: Introductionmentioning

confidence: 99%

“…The second author is grateful to the NSF for partial support under Grants DMS #1419448 and #1602070, and for a fellowship from the Sloan Foundation. 1 the Hilbert-Kunz multiplicity determines an upper semicontinuous R-valued function on ring spectra [Smi16].…”

Section: Introductionmentioning

confidence: 99%

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

Polstra

Tucker

2018

Alg. Number Th.

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We present a unified approach to the study of F -signature, Hilbert-Kunz multiplicity, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question 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.

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“…A simplified treatment of this proof, utilizing a lemma of Dutta, can be found in [12]. These techniques have inspired further results, e.g., concerning the upper semicontinuity of Hilbert-Kunz multiplicity [27], the lower semicontinuity of F -signature [24], and relations between Hilbert-Kunz multiplicity and F -signature [25].…”

Section: Introductionmentioning

confidence: 99%

Local Okounkov bodies and limits in prime characteristic

Hernández

Jeffries

2018

Math. Ann.

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This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call p-families of ideals, are ubiquitous in prime characteristic commutative algebra (e.g., they occur naturally in the theories of tight closure, Hilbert-Kunz multiplicity, and F -signature). We associate to each p-family of ideals an object in Euclidean space that is analogous to the Newton-Okounkov body of a graded family of ideals, which we call a p-body. Generalizing the methods used to establish volume formulas for the Hilbert-Kunz multiplicity and F -signature of semigroup rings, we relate the volume of a p-body to a certain asymptotic invariant determined by the corresponding p-family of ideals. We apply these methods to obtain new existence results for limits in positive characteristic, an analogue of the Brunn-Minkowski theorem for Hilbert-Kunz multiplicity, and a uniformity result concerning the positivity of a p-family.

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Upper semi-continuity of the Hilbert–Kunz multiplicity

Abstract: We prove that the Hilbert-Kunz multiplicity is upper semi-continuous in Ffinite rings and algebras of essentially finite type over an excellent local ring.

Cited by 17 publications

References 15 publications

Globalizing F-invariants

Globalizing F-invariants

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

Local Okounkov bodies and limits in prime characteristic

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