The Frobenius complexity of Hibi rings

Page, Janet

doi:10.1016/j.jpaa.2018.04.008

Cited by 10 publications

(12 citation statements)

References 14 publications

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“…As is noted in [Pag,Example 3.4], level property does not imply anticanonical level property nor anticanonical level property does not imply level property. Now we recall the definition of a Hibi ring.…”

Section: Posets and Hibi Ringsmentioning

confidence: 83%

“…As a special case, we see by Theorems 3.12, 5.1 and Corollary 4.8, the following fact whose anticanonical level part is [Pag,Theorem 4.6].…”

Section: Canonical and Anticanonical Analytic Spreadsmentioning

confidence: 92%

“…Definition 2.6 ( [Sta1,Pag]) Let R be a standard graded Cohen-Macaulay algebra over a field. If the degree of all the generators of the canonical module ω of R are the same, then we say that R is level.…”

Section: Posets and Hibi Ringsmentioning

confidence: 99%

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Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings

Miyazaki¹

2020

J. Math. Soc. Japan

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Let R K [H] be the Hibi ring over a field K on a finite distributive lattice H, P the set of join-irreducible elements of H and ω the canonical ideal of R K [H]. We show the powers ω (n) of ω in the group of divisors Div(R K [H]) is identical with the ordinary powers of ω, describe the K-vector space basis of ω (n) for n ∈ Z. Further, we show that the fiber cones n≥0 ω n /mω n and n≥0 (ω (−1) ) n /m(ω (−1) ) n of ω and ω (−1) are sum of the Ehrhart rings, defined by sequences of elements of P with a certain condition, which are polytopal complex version of Stanley-Reisner rings. Moreover, we show that the analytic spread of ω and ω (−1) are maximum of the dimensions of these Ehrhart rings. Using these facts, we show that the question of Page about Frobenius complexity is affirmative: lim p→∞ cx F (R K [H]) = dim( n≥0 ω (−n) /mω (−n) ) − 1, where p is the characteristic of the field K.

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Section: Posets and Hibi Ringsmentioning

confidence: 83%

“…As a special case, we see by Theorems 3.12, 5.1 and Corollary 4.8, the following fact whose anticanonical level part is [Pag,Theorem 4.6].…”

Section: Canonical and Anticanonical Analytic Spreadsmentioning

confidence: 92%

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Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings

Miyazaki¹

2020

J. Math. Soc. Japan

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“…We remark that here we are using the alternative definition of Frobenius complexity as in [13,Section 4], which was later adopted in [11,28]. If R is F-finite and complete, this definition coincides with the original one introduced by Enescu and Yao in [12].…”

Section: The Frobenius Complexitymentioning

confidence: 99%

“…In [11], the authors show that the Frobenius complexity is finite for standard graded rings over an F-finite field localized at the irrelevant maximal ideal. However, in general, cx F (R) is not known to be finite, except when the anticanonical cover is Noetherian [12, 4.7], when dim R 2 (in this case, cx F (R) 0) [12, 4.10], and in other particular cases [13,28].…”

Section: The Frobenius Complexitymentioning

confidence: 99%

Symbolic Analytic Spread: Upper Bounds and Applications

Dao¹,

Montaño²

2020

J. Inst. Math. Jussieu

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The symbolic analytic spread of an ideal $I$ is defined in terms of the rate of growth of the minimal number of generators of its symbolic powers. In this article, we find upper bounds for the symbolic analytic spread under certain conditions in terms of other invariants of $I$ . Our methods also work for more general systems of ideals. As applications, we provide bounds for the (local) Kodaira dimension of divisors, the arithmetic rank, and the Frobenius complexity. We also show sufficient conditions for an ideal to be a set-theoretic complete intersection.

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Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

2019

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The Frobenius complexity of Hibi rings

Abstract: We study the Frobenius complexity of Hibi rings over fields of characteristic p > 0. In particular, for a certain class of Hibi rings (which we call ω (−1) -level), we compute the limit of the Frobenius complexity as p → ∞.

Cited by 10 publications

References 14 publications

Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings

Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings

Symbolic Analytic Spread: Upper Bounds and Applications

Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

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