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.
Foundations for almost ring theory -- Release 7.5
4 Pith papers cite this work. Polarity classification is still indexing.
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abstract
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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background 2representative citing papers
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.
Establishes a representability criterion for v-sheaf modifications of formal schemes and applies it to parahoric level structures on local shtukas, yielding local representability of integral models of local Shimura varieties under hyperspecial levels.
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.
citing papers explorer
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A local-global correspondence for perfectoid purity
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.
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Algebraization of absolute perfectoidization via section rings
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.
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Relative representability and parahoric level structures
Establishes a representability criterion for v-sheaf modifications of formal schemes and applies it to parahoric level structures on local shtukas, yielding local representability of integral models of local Shimura varieties under hyperspecial levels.
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Flat Cohomological Purity for Syntomic Schemes over Valuation Rings
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.