Diagonal F-splitting and Symbolic Powers of Ideals

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.

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References in corpus (4)

• Frobenius Splitting in Commutative Algebra
• Symbolic powers of ideals
• Uniform Symbolic Topologies in Normal Toric Rings
• Symbolic and Ordinary Powers of Ideals in Hibi Rings