Hilbert-Kunz function and Hilbert-Kunz multiplicity of some ideals of the Rees algebra

Goel, Kriti; Koley, Mitra; Verma, J. K.

doi:10.48550/arxiv.1911.03889

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Section: Introductionmentioning

confidence: 99%

“…The Hilbert-Kunz functions of the Rees algebra, associated graded ring and the extended Rees algebra have been studied by K. Eto and K.-i. Yoshida in [3] and by K. Goel, M. Koley and J. K. Verma in [5].In order to recall one of the main results of Eto and Yoshida, we set up some notation first. Let (R, m) be a Noetherian local ring of dimension d and of prime characteristic p. Let q = p e where e is a non-negative integer.…”

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Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex

Banerjee¹,

Goel²,

Verma³

2020

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Let R be the face ring of a simplicial complex of dimension d − 1 and R(n) be the Rees algebra of the maximal homogeneous ideal n of R. We show that the generalized Hilbert-Kunz function HK(s) = (R(n)/(n, nt) [s] ) is given by a polynomial for all large s. We calculate it in many examples and also provide a Macaulay2 code for computing HK(s).Dedicated to Roger Wiegand and Silvia Wiegand on the occasion of their 150 th birthday IntroductionThe objective of this paper is to find the generalized Hilbert-Kunz function of the maximal homogeneous ideal of the Rees algebra of the maximal homogeneous ideal of the face ring of a simplicial complex. The Hilbert-Kunz functions of the Rees algebra, associated graded ring and the extended Rees algebra have been studied by K. Eto and K.-i. Yoshida in [3] and by K. Goel, M. Koley and J. K. Verma in [5].In order to recall one of the main results of Eto and Yoshida, we set up some notation first. Let (R, m) be a Noetherian local ring of dimension d and of prime characteristic p. Let q = p e where e is a non-negative integer. The q th Frobenius power of an ideal I is defined to be I [q] = (a q | a ∈ I). Let I be an m-primary ideal. The Hilbert-Kunz function of I is the function HK I (q) = (R/I [q] ). This function, for I = m, was introduced by E. Kunz in [8] who used it to characterize regular local rings.The Hilbert-Kunz multiplicity of an m-primary ideal I is defined as e HK (I) = lim q→∞ (R/I [q] )/q d . It was introduced by P. Monsky in [10]. We refer the reader to an excellent survey paper of C. Huneke [7] for further details. Eto and Yoshida calculated the Hilbert-Kunz multiplicity of various blowup algebras of an ideal under certain conditions. Put c(d) = (d/2) + d/(d + 1)!. They proved the following. Theorem 1.1. Let (R, m) be a Noetherian local ring of prime characteristic p > 0 with d = dim R ≥ 1. Then for any m-primary ideal I, we have e HK (R(I)) ≤ c(d) • e(I).

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%