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July, 2020 Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings
Mitsuhiro MIYAZAKI
J. Math. Soc. Japan 72(3): 991-1023 (July, 2020). DOI: 10.2969/jmsj/81418141

Abstract

Let ${\mathcal R}_\mathbb{K}[H]$ be the Hibi ring over a field $\mathbb{K}$ on a finite distributive lattice $H$, $P$ the set of join-irreducible elements of $H$ and $\omega$ the canonical ideal of ${\mathcal R}_\mathbb{K}$[H]. We show the powers $\omega^{(n)}$ of $\omega$ in the group of divisors $\mathrm{Div}({\mathcal R}_\mathbb{K}[H])$ is identical with the ordinary powers of $\omega$, describe the $\mathbb{K}$-vector space basis of $\omega^{(n)}$ for $n \in \mathbb{Z}$. Further, we show that the fiber cones $\bigoplus_{n \geq 0} \omega^n/\mathfrak{m} \omega^n$ and $\bigoplus_{n \geq 0} (\omega^{(-1)})^n/\mathfrak{m} (\omega^{(-1)})^n$ of $\omega$ and $\omega^{(-1)}$ 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 $\omega$ and $\omega^{(-1)}$ are maximum of the dimensions of these Ehrhart rings. Using these facts, we show that the question of Page about Frobenius complexity is affirmative: $\lim_{p \to \infty} \mathrm{cx}_F ({\mathcal R}_\mathbb{K}[H]) = \dim(\bigoplus_{n \geq 0} \omega^{(-n)}/\mathfrak{m} \omega^{(-n)}) - 1$, where $p$ is the characteristic of the field $\mathbb{K}$.

Funding Statement

The author is supported partially by JSPS KAKENHI Grant Number JP15K04818.

Citation

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Mitsuhiro MIYAZAKI. "Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings." J. Math. Soc. Japan 72 (3) 991 - 1023, July, 2020. https://doi.org/10.2969/jmsj/81418141

Information

Received: 4 October 2018; Revised: 8 April 2019; Published: July, 2020
First available in Project Euclid: 28 January 2020

zbMATH: 07257219
MathSciNet: MR4125854
Digital Object Identifier: 10.2969/jmsj/81418141

Subjects:
Primary: 13A35
Secondary: 06A07, 13A30, 13F50

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 3 • July, 2020
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