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Perfectoid pure singularities

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arxiv 2409.17965 v2 pith:XHKYGHOF submitted 2024-09-26 math.AG math.ACmath.NT

Bhargav Bhatt , Linquan Ma , Zsolt Patakfalvi , Karl Schwede , Kevin Tucker , Joe Waldron , Jakub Witaszek This is my paper · ORCID
classification math.AGmath.ACmath.NT
keywords puresingularitiesconditionperfectoidboiscallnotionsprove
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abstract

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 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 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 \claperfdpure 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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Brian\c{c}on-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings

    math.AC 2025-10 conditional novelty 8.0 of 10

    For pseudo-rational and many Du Bois singularities, the full Briançon–Skoda containment J^{n+k-1} ⊆ J^k holds, and quasi-excellent finite-dimensional rings satisfy uniform Briançon–Skoda and uniform Artin–Rees.

  2. Algebraization of absolute perfectoidization via section rings

    math.AG 2026-04 unverdicted novelty 7.0 of 10

    Graded absolute perfectoidization of G-graded adic rings yields an algebraization of the structure sheaf of projective-type formal schemes.

  3. Quasi-canonical lifting of projective varieties in positive characteristic

    math.AG 2025-06 conditional novelty 7.0 of 10

    A descent theorem: a finite étale cover with a quasi-canonical lifting, of degree prime to p, forces the base variety to have a canonical lifting over the Witt vectors.

  4. {\delta}-rings, perfectoid towers, and lim Cohen-Macaulay sequences

    math.AC 2025-09 conditional novelty 6.0 of 10

    Perfectoid towers over complete local domains yield lim Cohen-Macaulay sequences, and new perfectoid towers are constructed, including the first with p-torsion.

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