Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

Watanabe, Kei-ichi; Yoshida, Ken-ichi

doi:10.1006/jabr.1999.7956

Cited by 90 publications

(86 citation statements)

References 19 publications

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“…In [4] Hanes recently showed that, in fact, e(I)/d! < e HK (I), for every m-primary ideal I, answering affirmatively a question raised by Watanabe and Yoshida (Question 2.9 in [7]). …”

Section: Lower Bounds On the Hilbert-kunz Multiplicitysupporting

confidence: 52%

“…In the one-dimensional case this is trivial, HK (1) = 1. Work of Watanabe and Yoshida [7], together with our Remark 2.6, shows that HK (2) = 1/2. Furthermore, Watanabe and Yoshida give a classification of all twodimensional Cohen-Macaulay rings with Hilbert-Kunz multiplicity less than 2.…”

Section: Proposed Problemsmentioning

confidence: 65%

“…Definition 2.1. Let I be an m-primary ideal in R. The Hilbert-Samuel multiplicity of I with respect to R is defined by For a collection of basic results regarding these notions, we refer the reader to [5], or [7]. We list here only a few well-known facts that will be needed later.…”

Section: Lower Bounds On the Hilbert-kunz Multiplicitymentioning

confidence: 99%

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On rings with small Hilbert-Kunz multiplicity

Blickle

Enescu

2004

Proc. Amer. Math. Soc.

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Abstract. A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed p and d, there exists a number (d, p) > 0 such that for any nonregular unmixed ring R its Hilbert-Kunz multiplicity is at least 1 + (d, p). We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and F -rational.

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“…In [4] Hanes recently showed that, in fact, e(I)/d! < e HK (I), for every m-primary ideal I, answering affirmatively a question raised by Watanabe and Yoshida (Question 2.9 in [7]). …”

Section: Lower Bounds On the Hilbert-kunz Multiplicitysupporting

confidence: 52%

Section: Proposed Problemsmentioning

confidence: 65%

See 1 more Smart Citation

On rings with small Hilbert-Kunz multiplicity

Blickle

Enescu

2004

Proc. Amer. Math. Soc.

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“…(Fundamental properties (cf. [13], [15], [16], [18], [22])) Let (A, m, k) be a local ring of positive characteristic. Let I, I be m-primary ideals of A, and let M be a finitely generated A-module.…”

Section: Introductionmentioning

confidence: 99%

Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings

Watanabe

Yoshida

2005

Nagoya Mathematical Journal

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Abstract. In this paper, we investigate the lower bound sHK(p, d) of HilbertKunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on threedimensional local rings. In fact, as a main result, we will prove that sHK(p, 3) = 4/3 and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degenerate quadric hypersurfaceFurthermore, we pose a generalization of the main theorem to the case of dim A ≥ 4 as a conjecture, and show that it is also true in case dim A = 4 using the similar method as in the proof of the main theorem.

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“…They can be found in [WY,Proposition 4.1,Lemma 4.2] in slightly different forms. The filtration argument used in the proof can also be found in [Ha,Proposition 5.2.1].…”

Section: Preliminariesmentioning

confidence: 99%