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

Banerjee, Arindam; Goel, Kriti; Verma, J. K.

doi:10.1090/conm/773/15543

Cited by 2 publications

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“…)) is the face ring of ∆. Observe that R is a 2-dimensional Cohen-Macaulay ring with f -vector f (∆) = (1,4,4). Let n = (x 1 , x 2 , x 3 , x 4 ) denote the maximal ideal of R. Using Theorem 3.4, it follows that 4 = e 0 (n) = e HK (n) and lim q→∞ e 1 (n [q] )/q 2 = 0.…”

Section: Counter-example To Smirnov's Conjecturementioning

confidence: 99%

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On the Hilbert-Samuel coefficients of Frobenius powers of an ideal

Banerjee¹,

Goel²,

Verma³

2022

Preprint

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We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an m-primary ideal exists in a Noetherian local ring (R, m) with prime characteristic p > 0. This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring R of a simplicial complex and an ideal J generated by pure powers of the variables, the generalized Hilbert-Kunz function ℓ(R/(J [q] ) k ) is a polynomial for all q, k and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of J in terms of Hilbert-Samuel multiplicity of J. We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal.

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Section: Counter-example To Smirnov's Conjecturementioning

confidence: 99%

“…(1) Let R be the face ring of the simplicial complex on r vertices, r ≥ 3, as inExample 3By putting q = 1 in (3.4), we get for all k ≥ 1,ℓ(R/n k ) = (r − 1) n) − ℓ(R/n) = (r − 1) − r − 1 − (r − 2) = r − 2 = e 1 (n),…”

mentioning

confidence: 99%

On the Hilbert-Samuel coefficients of Frobenius powers of an ideal

Banerjee¹,

Goel²,

Verma³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

On the Hilbert-Samuel coefficients of Frobenius powers of an ideal

Banerjee,

Goel,

Verma

2024

Proc. Amer. Math. Soc.

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We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an m \mathfrak {m} -primary ideal exists in a Noetherian local ring ( R , m ) (R,\mathfrak {m}) with prime characteristic p > 0. p>0. This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring R R of a simplicial complex and an ideal J J generated by pure powers of the variables, the generalized Hilbert-Kunz function ℓ ( R / ( J [ q ] ) k ) \ell (R/(J^{[q]})^k) is a polynomial for all q , k q,k and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of J J in terms of Hilbert-Samuel multiplicity of J . J. We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal.

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

Abstract: Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}({\mathfrak {n}}) be the Rees algebra of the maximal homogeneous ideal n {\mathfrak {n}} of R . R. We show that the generalized Hilbert-Kunz function … Show more

Cited by 2 publications

References 9 publications

On the Hilbert-Samuel coefficients of Frobenius powers of an ideal

On the Hilbert-Samuel coefficients of Frobenius powers of an ideal

On the Hilbert-Samuel coefficients of Frobenius powers of an ideal

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