Some surprising Hilbert-Kunz functions

In this paper, we study local rings of small Hilbert-Kunz multiplicity. In particular, we prove that an unmixed local ring of Hilbert-Kunz multiplicity one is regular and classify two-dimensional Cohen-Macaulay local rings whose Hilbert-Kunz multiplicity is 2 or less. Also, we investigate the inequality between the multiplicity and the colength of the tight closure of parameter ideals inverse to the usual inequality between multiplicity and colength.

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“…Remark. The above theorem has been proved by Han and Monsky [HM,(5.8),(5.9)] in the case of complete intersections.…”

Section: Thus It Is Natural To Askmentioning

confidence: 62%

“…To conclude this section, we propose the following conjecture, which holds if A is a complete intersection local ring; see also Proposition 2.13, Han and Monsky [HM,(5.8)(5.9)] and Dutta [Du]. Note that the converse of (2) is not true in general.…”

Section: Fundamental Properties Of Hilbert-kunz Multiplicitymentioning

confidence: 90%

Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

Watanabe

Yoshida

2000

Journal of Algebra

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“…If A is isomorphic to A p,4 , then by [10], it is known that e HK (A) = 29p 2 +15 24p 2 +12 . Suppose that A is not isomorphic to A p,4 .…”

Section: Also the Equality Holds If And Only Ifmentioning

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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“…(A consequence of this would be that each function can be described by a finite set of functional equations of a simple type, and a finite set of initial values.) One result is implicit in [2] and [3]-if r = s + 1, u i = x i and u s+1 = s i=1 x i , then ϕ u is a p-fractal. In this paper we answer (1) and (2) in the simplest non-trivial cases, proving the following theorems.…”

Section: Introductionmentioning

confidence: 99%

p-Fractals and power series–I. Some 2 variable results

Monsky

Teixeira

2004

Journal of Algebra

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u 1 , . . . , u r are in k[[x 1 , . . . , x s ]] with k and deg(u 1 , . . . , u r ) finite. Intending applications to Hilbert-Kunz theory, we code the numbers deg(u a 1 1 , . . . , u a r r ) into a function ϕ u , which empirically satisfies many functional equations related to "magnification by p," where p = char k. p-fractals, introduced here, formalize these ideas.In the first interesting case (r = 3, s = 2), the ϕ u are p-fractals. Our proof uses functions ϕ I attached to ideals I and square-free elements h of A = k [[x, y]]. The finiteness of the set of ideal classes in A/(h) and the existence of "magnification maps" on this set show the ϕ I to be p-fractals.We describe further functional equations coming from a theory of reflection maps on ideal classes, and the paper concludes with examples and open questions.

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Some surprising Hilbert-Kunz functions

Cited by 70 publications

References 2 publications

Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

Hilbert–Kunz Multiplicity and an Inequality between Multiplicity and Colength

Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings

p-Fractals and power series–I. Some 2 variable results

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