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arxiv: math/0409584 · v13 · submitted 2004-09-29 · 🧮 math.AG · math.NT

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Foundations for almost ring theory -- Release 7.5

Ofer Gabber , Lorenzo Ramero
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classification 🧮 math.AG math.NT
keywords almostproofreleasefoundationsgeneralizationperfectoidspacesandr
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This is release 7.5 of our project, aiming to provide a complete treatment of the foundations of almost ring theory, following and extending Faltings's method of "almost etale extensions". The central result is the "almost purity theorem", for whose proof we adapt Scholze's method, based on his perfectoid spaces. This release provides the foundations for our generalization of Scholze's perfectoid spaces, and reduces the proof of the almost purity theorem to a general assertion concerning the \'etale topology of adic spaces, whose proof uses previous work by the first author. As usual, this new release is a mix of corrections and various improvements, with a final chapter dedicated to applications; notably, we include a generalization of Y.Andr\'e's "perfectoid Abhyankar's lemma" which we use to give a proof of a generalization of the "direct summand conjecture", extending Andr\'e's recent work.

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Cited by 4 Pith papers

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

  1. A local-global correspondence for perfectoid purity

    math.AG 2026-04 unverdicted novelty 7.0

    A correspondence is shown between lim-perfectoid splitting of projective schemes and lim-perfectoid purity of their Gorenstein section rings, supplying new examples of lim-perfectoid pure rings.

  2. Algebraization of absolute perfectoidization via section rings

    math.AG 2026-04 unverdicted novelty 7.0

    A graded absolute perfectoidization is built for G-graded adic rings, with the key result that the absolute perfectoidization of the structure sheaf on projective-type formal schemes algebraizes.

  3. Flat Cohomological Purity for Syntomic Schemes over Valuation Rings

    math.AG 2026-05 unverdicted novelty 6.0

    Proves that cohomology of syntomic schemes over valuation rings is unchanged by removing closed subschemes of suitable fibrewise codimension, extending Česnavičius–Scholze to non-noetherian cases.

  4. A local-global correspondence for perfectoid purity

    math.AG 2026-04 unverdicted novelty 5.0

    A correspondence links lim-perfectoid splitting of projective schemes to lim-perfectoid purity of their Gorenstein section rings, supplying new examples of lim-perfectoid pure rings beyond complete intersections and s...