Symbolic and Ordinary Powers of Ideals in Hibi Rings

Abstract

We exhibit a class of Hibi rings which are diagonally F-regular over fields of positive characteristic, and diagonally $F$-regular type over fields of characteristic zero, in the sense of Carvajal-Rojas and Smolkin. It follows that such Hibi rings satisfy the uniform symbolic topology property effectively in all characteristics. Namely, for rings $R$ in this class of Hibi rings, we have $P^{(dn)}\subseteq P^n$ for all $P\in \mathrm{Spec}R$, where $d=\dim (R)$. Further, we demonstrate that all Hibi rings over fields of positive characteristic are 2-diagonally $F$-regular, and that the simplest Hibi ring not contained in the above class is not 3-diagonally $F$-regular in any characteristic. The former implies that $P^{(2d)}\subseteq P^2$ for all $P\in \mathrm{Spec}R$.

15 pages, plus an appendix. Comments welcome