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
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
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