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

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 ordinal 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 ${⨁}_{n\ge 0}{\omega }^n/\mathfrak{m}{\omega }^n$ and ${⨁}_{n\ge 0}({\omega }^{(-1)}{)}^n/\mathfrak{m}({\omega }^{(-1)}{)}^n$ 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: ${\lim }_{p\to \infty }{\mathrm{cx}}_F({\mathcal{R}}_{\mathbb{K}}[H])=\dim ({⨁}_{n\ge 0}{\omega }^{(-n)}/\mathfrak{m}{\omega }^{(-n)})-1$, where $p$ is the characteristic of the field $\mathbb{K}$.

Minor changes, including change of the title, correction of errors and adding some examples

References in corpus (1)

• The Frobenius complexity of Hibi rings

Cited by in corpus (2)

• Non-Gorenstein loci of Ehrhart rings of chain and order polytopes
• On the canonical ideal of the Ehrhart ring of the chain polytope of a poset