Volume 8

Eighth volume of Épijournal de Géométrie Algébrique - 2024

1. Diagonal F-splitting and Symbolic Powers of Ideals

Daniel Smolkin.
Let $J$ be any ideal in a strongly $F$-regular, diagonally $F$-split ring $R$essentially of finite type over an $F$-finite field. We show that $J^{s+t}\subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon})$ for all $s, t, \epsilon> 0$ for which the formula makes sense. We use this to show a number of novelcontainments between symbolic and ordinary powers of prime ideals in thissetting, which includes all determinantal rings and a large class of toricrings in positive characteristic. In particular, we show that $P^{(2hn)}\subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.