Daniel Smolkin - Diagonal F-splitting and Symbolic Powers of Ideals

epiga:9918 - Épijournal de Géométrie Algébrique, January 22, 2024, Volume 8 - https://doi.org/10.46298/epiga.2023.9918
Diagonal F-splitting and Symbolic Powers of IdealsArticle

Authors: 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 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.

    Volume: Volume 8
    Published on: January 22, 2024
    Accepted on: October 11, 2023
    Submitted on: August 15, 2022
    Keywords: Mathematics - Commutative Algebra,Mathematics - Algebraic Geometry,13A15 (secondary), 13A35 (primary), 14B05

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