Introduces Higgs bundles on the Fargues-Fontaine curve, establishes a BNR correspondence, and shows an injective étale-stack map from B_dR^+-affine Springer fibers to the Hitchin fiber inducing category equivalence on geometric points.
Foundations for almost ring theory -- Release 7.5
7 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
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
Program to prove categorical local Langlands conjecture via compatibility hypothesis for GL_n, induction on Levi subgroups for general groups, plus new spectral finiteness, duality, and admissible ind-coherent sheaf results.
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.
A Lean formalization of q-Pochhammer symbols, Bailey pairs and related primitives yields fully verified proofs of the Jacobi triple product and Rogers–Ramanujan identities over strongly non-archimedean rings.
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.
Graded absolute perfectoidization of G-graded adic rings yields an algebraization of the structure sheaf of projective-type formal schemes.
citing papers explorer
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Higgs bundles on the Fargues-Fontaine curve
Introduces Higgs bundles on the Fargues-Fontaine curve, establishes a BNR correspondence, and shows an injective étale-stack map from B_dR^+-affine Springer fibers to the Hitchin fiber inducing category equivalence on geometric points.
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The categorical local Langlands conjecture
Program to prove categorical local Langlands conjecture via compatibility hypothesis for GL_n, induction on Levi subgroups for general groups, plus new spectral finiteness, duality, and admissible ind-coherent sheaf results.
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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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Formalized $q$-series: The Rogers-Ramanujan Identities and Beyond
A Lean formalization of q-Pochhammer symbols, Bailey pairs and related primitives yields fully verified proofs of the Jacobi triple product and Rogers–Ramanujan identities over strongly non-archimedean rings.
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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.
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Algebraization of absolute perfectoidization via section rings
Graded absolute perfectoidization of G-graded adic rings yields an algebraization of the structure sheaf of projective-type formal schemes.