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

Abstract

Let ${\cal R}_{\mathbb{K}}[H]$ 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 $ω$ the canonical ideal of ${\cal R}_{\mathbb{K}}[H]$. We show the powers $ω^{(n)}$ of $ω$ in the group of divisors $\mathrm{Div}({\cal R}_{\mathbb{K}}[H])$ is identical with the ordinal powers of $ω$, describe the $\mathbb{K}$-vector space basis of $ω^{(n)}$ for $n\in\mathbb{Z}$. Further, we show that the fiber cones $\bigoplus_{n\geq 0}ω^n/\mathfrak{m}ω^n$ and $\bigoplus_{n\geq0}(ω^{(-1)})^n/\mathfrak{m}(ω^{(-1)})^n$ 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: $\lim_{p\to\infty}\mathrm{cx}_F({\cal R}_{\mathbb{K}}[H])= \dim(\bigoplus_{n\geq0}ω^{(-n)}/\mathfrak{m}ω^{(-n)})-1$, where $p$ is the characteristic of the field $\mathbb{K}$.