Diagonal F-splitting and Symbolic Powers of Ideals

TL;DR

This paper establishes new containments between symbolic and ordinary powers of ideals in strongly F-regular, diagonally F-split rings, including determinantal and toric rings, advancing understanding of ideal powers in positive characteristic.

Contribution

It introduces a novel diagonal F-splitting technique to derive containments between symbolic and ordinary powers of ideals in specific algebraic rings.

Findings

Proves $J^{s+t} subseteq au(J^{s - ext{epsilon}}) au(J^{t- ext{epsilon}})$ in certain rings.

Shows $P^{(2hn)} subseteq P^n$ for prime ideals in these rings.

Includes all determinantal and a large class of toric rings in the results.

Abstract

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 novel containments between symbolic and ordinary powers of prime ideals in this setting, which includes all determinantal rings and a large class of toric rings in positive characteristic. In particular, we show that $P^{(2hn)} \subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.