Perfectoid pure singularities
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910621313794048 |
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| author | Bhatt, Bhargav Ma, Linquan Patakfalvi, Zsolt Schwede, Karl Tucker, Kevin Waldron, Joe Witaszek, Jakub |
| author_facet | Bhatt, Bhargav Ma, Linquan Patakfalvi, Zsolt Schwede, Karl Tucker, Kevin Waldron, Joe Witaszek, Jakub |
| contents | Fix a prime number $p$. Inspired by the notion of $F$-pure or $F$-split singularities, we study the condition that a Noetherian ring with $p$ in its Jacobson radical is pure inside some perfectoid (classical) ring, a condition we call \emph{perfectoid pure}. We also study a related a priori weaker condition which asks that $R$ is pure in its absolute perfectoidization, a condition we call \emph{lim-perfectoid pure}. We show that both these notions coincide when $R$ is LCI. Mixed characteristic analogs of $F$-injective and Du Bois singularities are also explored. We study these notions of singularity, proving that they are weakly normal and that they are Du Bois after inverting $p$. We also explore the behavior of perfectoid singularities under finite covers and their relation to log canonical singularities. Finally, we prove an inversion of adjunction result in the LCI setting, and use it to prove that many common examples are perfectoid pure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perfectoid pure singularities Bhatt, Bhargav Ma, Linquan Patakfalvi, Zsolt Schwede, Karl Tucker, Kevin Waldron, Joe Witaszek, Jakub Algebraic Geometry Commutative Algebra Number Theory 14G45, 14F18, 14B05, 13A35, 11G25 Fix a prime number $p$. Inspired by the notion of $F$-pure or $F$-split singularities, we study the condition that a Noetherian ring with $p$ in its Jacobson radical is pure inside some perfectoid (classical) ring, a condition we call \emph{perfectoid pure}. We also study a related a priori weaker condition which asks that $R$ is pure in its absolute perfectoidization, a condition we call \emph{lim-perfectoid pure}. We show that both these notions coincide when $R$ is LCI. Mixed characteristic analogs of $F$-injective and Du Bois singularities are also explored. We study these notions of singularity, proving that they are weakly normal and that they are Du Bois after inverting $p$. We also explore the behavior of perfectoid singularities under finite covers and their relation to log canonical singularities. Finally, we prove an inversion of adjunction result in the LCI setting, and use it to prove that many common examples are perfectoid pure. |
| title | Perfectoid pure singularities |
| topic | Algebraic Geometry Commutative Algebra Number Theory 14G45, 14F18, 14B05, 13A35, 11G25 |
| url | https://arxiv.org/abs/2409.17965 |