Geodesic
Diagonal F-splitting and Symbolic Powers of Ideals
Épijournal de Géométrie Algébrique, Tome 8 (2024)

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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.
DOI : 10.46298/epiga.2023.9918
Classification : 13A15, 13A35, 14B05 
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     author = {Smolkin, Daniel},
     title = {Diagonal {F-splitting} and {Symbolic} {Powers} of {Ideals}},
     journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
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     year = {2024},
     doi = {10.46298/epiga.2023.9918},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2023.9918/}
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Smolkin, Daniel. Diagonal F-splitting and Symbolic Powers of Ideals. Épijournal de Géométrie Algébrique, Tome 8 (2024). doi: 10.46298/epiga.2023.9918

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