Local Okounkov bodies and limits in prime characteristic

Hernández, Daniel J.; Jeffries, Jack

doi:10.1007/s00208-018-1651-6

Cited by 8 publications

(3 citation statements)

References 43 publications

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“…This is done by using the length of the local cohomology at the maximal ideal. This notion, known as the generalized Hilbert-Kunz multiplicity, was proved to exist and further developed by Epstein and Yao [38], Hernández and Jeffries [50], and Dao and Smirnov [34]. It should be noted that this is different from another generalized version for monomial ideals, primary to a maximal ideal, considered by Conca,Miller,Robinson and Swanson [30,70,89] (see Subsection 3.6).…”

Section: History In Briefmentioning

confidence: 99%

The Shape of Hilbert-Kunz Functions

Chan¹

2021

Preprint

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We discuss Hilbert-Kunz function from when it was originally defined to its recent developments. A brief history of Hilbert-Kunz theory is first recounted. Then we review several techniques involved in the study of Hilbert-Kunz functions by presenting some illustrative proofs without going into details of the technicalities.The second part of this article focuses on the Hilbert-Kunz function of an affine normal semigroup ring and relates it to Ehrhart quasipolynomials. We pay extra attention to its periodic behavior and discuss how the cellular decomposition constructed by Bruns and Gubeladze fits into the computation of the functions. The closed forms of the Hilbert-Kunz function of some examples are presented. The discussion in this part highlights the close relationship between Hilbert-Kunz function and Ehrhart theory.

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Section: History In Briefmentioning

confidence: 99%

The Shape of Hilbert-Kunz Functions

Chan¹

2021

Preprint

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“…This setting has recently been analyzed in detail in [15]. The connections of these limits to tight closure, invariants of vector bundles, and volumes of certain convex bodies have been explored in [3,4,15,16,19] In light of Theorem 5.3, the previous question reduces to the following.…”

Section: Related Work and Open Questionsmentioning

confidence: 99%

Length of local cohomology of powers of ideals

Dao

Montaño

2018

Trans. Amer. Math. Soc.

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Let R be a polynomial ring over a field k with irrelevant ideal m and dimension d. Let I be a homogeneous ideal in R. We study the asymptotic behavior of the length of the modules H i m (R/I n ) for n ≫ 0. We show that for a fixed number α ∈ Z, lim sup n→∞< ∞ when X = Proj R/I is locally a complete intersection (lci) and i dim X. We also establish that the actual limit exists and is rational for certain classes of monomial ideals I such that the lengths of local cohomology of I n are eventually finite. Our proofs use Gröbner deformation and Presburger arithmetic. Finally, we utilize more traditional commutative algebra techniques to show that lim inf n→∞ λ(H i m (R/I n )) n d > 0 under certain conditions when R/I is either F -pure or lci.2010 Mathematics Subject Classification. 13D45, 13A30, 14B05, 05E40.

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“…We will then use that equivalence to compute the s-multiplicity for a few toric rings. See [HJ17] for a more general treatment of the correspondence between limits in positive characteristic and volumes in real space. Definition 5.1.…”

Section: S-multiplicity Of Toric Ringsmentioning

confidence: 99%

Interpolating between Hilbert–Samuel and Hilbert–Kunz multiplicity

Taylor

2018

Journal of Algebra

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We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real space. 1 J (p e ) = ∞ for all e and so c J (I) = ∞. If J = R then ν I J (p e ) = −∞ for all e and so c J (I) = −∞. Suppose 1 ⊆ √ J = R, so that for all e, 0 ≤ ν I J (p e ) and so c J (I) ≥ 0. That c J (I) exists in the case I ⊆ √ J is [DNBP16, Theorem 3.4], the proof of which also shows that c J (I) < ∞ in this case. If J ⊆ √ I, then µ I J (p e ) = 1 for all e and so b J (I) = 0. If I = R, then µ I J (p e ) = ∞ for all e, and so b J (I) = ∞. Suppose that J ⊆ √ I = R. The proof of the existence of b J (I) in this case is nearly identical to that of the existence of c J (I). Let e, e ′ ∈ N. We have that J [p e+e ′ ] = J [p e ′ ] [p e ] ⊆ I µ I J (p e ′ )−1 [p e ]

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Local Okounkov bodies and limits in prime characteristic

Cited by 8 publications

References 43 publications

The Shape of Hilbert-Kunz Functions

The Shape of Hilbert-Kunz Functions

Length of local cohomology of powers of ideals

Interpolating between Hilbert–Samuel and Hilbert–Kunz multiplicity

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