The Hilbert-Kunz function

Monsky, Paul

doi:10.1007/bf01457082

Cited by 193 publications

(182 citation statements)

References 2 publications

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“…In general there is an exact sequence 0 → T → M → M → 0 with T torsion and M torsion free. The exact sequence Tor [Mo1] shows that e n (T ) = cq d−1 + O(q d−2 ) for some c ≥ 0, and the result for M yields the result for M .…”

Section: Theorem 112 Let (R M K) Be An Excellent Local Normal Rmentioning

confidence: 57%

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Hilbert-Kunz Functions for Normal Rings

Huneke

McDermott

Monsky

2004

Mathematical Research Letters

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Section: Theorem 112 Let (R M K) Be An Excellent Local Normal Rmentioning

confidence: 57%

“…The basic question this paper studies is how e n (M ) depends on n. The results of [Mo1] show that e n (M ) = αq d + O(q d−1 ) for some real α. In section 1 we strengthen this proving: In the situation of Theorem 1 it sometimes happens that β(M ) = 0 provided that M = R (or more generally when M is torsion-free).…”

Section: Introductionmentioning

confidence: 99%

Hilbert-Kunz Functions for Normal Rings

Huneke

McDermott

Monsky

2004

Mathematical Research Letters

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“…[q] ) = e HK (R)q d + O(q d−1 ) as was first shown by P. Monsky [Mon83] building on work of E. Kunz. Much subtly lies in the lower order terms.…”

Section: Introductionmentioning

confidence: 78%

The 𝑠-multiplicity function of 2×2-determinantal rings

Miller

Taylor

2018

Proc. Amer. Math. Soc.

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Abstract. This article generalizes joint work of the first author and I. Swanson to the s-multiplicity recently introduced by the second author. For k a field and X = [x i,j ] a m × n-matrix of variables, we utilize Gröbner bases to give a closed form the length, q is a sufficiently large power of p, and m is the homogeneous maximal ideal of k [X]. This shows this length is always eventually a polynomial function of q for all s.

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“…We propose a characteristic-free interpretation of definition of the Hilbert-Kunz multiplicity, see [37], in terms of local entropy. From Theorem 2 it quickly follows that the Hilbert-Kunz multiplicity of a regular local ring with respect to any endomorphism of finite length is 1.…”

Section: Definitionmentioning

confidence: 99%

Entropy and flatness in local algebraic dynamics

Majidi-Zolbanin¹,

Miasnikov²,

Szpiro³

2013

Publicacions Matemàtiques

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For a local endomorphism of a noetherian local ring we introduce a notion of entropy, along with two other asymptotic invariants. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that every finite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to a finite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a point fixed by a finite self-morphism inherits a local endomorphism whose entropy is well-defined. This situation arises at the vertex of the affine cone over a projective variety with a polarized self-morphism, where we compare entropy with degree.

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The Hilbert-Kunz function

Cited by 193 publications

References 2 publications

Hilbert-Kunz Functions for Normal Rings

Hilbert-Kunz Functions for Normal Rings

The 𝑠-multiplicity function of 2×2-determinantal rings

Entropy and flatness in local algebraic dynamics

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