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 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 $\mathbb{K}$.