Diagonal F-splitting and Symbolic Powers of Ideals
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866916047536259072 |
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| author | Smolkin, Daniel |
| author_facet | Smolkin, Daniel |
| contents | 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 $J^{s+t} \subseteq τ(J^{s - ε}) τ(J^{t-ε})$ for all $s, t, ε> 0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_00051 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Diagonal F-splitting and Symbolic Powers of Ideals Smolkin, Daniel Commutative Algebra Algebraic Geometry 13A15 (secondary), 13A35 (primary), 14B05 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 $J^{s+t} \subseteq τ(J^{s - ε}) τ(J^{t-ε})$ for all $s, t, ε> 0$ 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. |
| title | Diagonal F-splitting and Symbolic Powers of Ideals |
| topic | Commutative Algebra Algebraic Geometry 13A15 (secondary), 13A35 (primary), 14B05 |
| url | https://arxiv.org/abs/2208.00051 |