Drinfeld's Lemma for Perfectoid Spaces and Overconvergence of Multivariate $(\phi, \Gamma)$-Modules

Carter, Annie; Kedlaya, Kiran S.; Zábrádi, Gergely

doi:10.25537/dm.2021v26.1329-1393

Cited by 2 publications

References 17 publications

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Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

Ray¹,

Wei²,

Zábrádi³

2021

Bul. Soc. Math. France

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Let K be a complete discretely valued field with mixed characteristic (0, p) and imperfect residue field k α . Let ∆ be a finite set. We construct an equivalence of categories between finite dimensional F p -representations of the product of ∆ copies of the absolute Galois group of K and multivariable étale (ϕ, Γ)-modules over a multivariable Laurent series ring over k α .Résumé ((ϕ, Γ)-modules multivariables et représentations du produit du groupe de Galois: le cas des corps résiduels imparfaits) Soit K un corps discrètement valué à charactéristique mixte (0, p) et un corps résiduel imparfait k α . Soit ∆ un ensemble fini. Nous établissons une équivalence de catégories entre des représentations de dimensions finies sur F p du produit de ∆ copies du groupe absolu de Galois de K et des (ϕ, Γ)-modules étales multivariables sur un anneau multivariable des séries Laurent sur k α .

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Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

Ray¹,

Wei²,

Zábrádi³

2021

Bul. Soc. Math. France

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Analytic Geometry and Hodge-Frobenius Structure Continued

Tong¹

2020

Preprint

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This is our sequel to our previous work on the corresponding generalized Frobenius modules over some big multivariate Robba rings. We will go beyond our previous discussion where we focused on the corresponding analytic functions on polydiscs and polyannuli in the strictly affinoid situation, and general Hodge-Frobenius structures which are admissible in the corresponding context in our previous work.However we will in this paper to consider some more complicated version of the rings. We will use some partial Frobenius to perfectize partially the rings defined above. Therefore we will in some more uniform way to denote the rings in the following different way:Π an,r I ,I,I,∅,A := Π an,r I ,I,A (2.2) Π an,con,I,I,∅,A := Π an,con,I,∅,A .(2.3) Now we follow the idea in [Ked2, Definition 5.2.1] to define some extended version of the rings. We will have the following rings to be: Π [s I ,r I ],I, J,I\J,A , (2.4) Π [s I ,r I ],I, J,I\J,A , (2.5) Π [s I ,r I ],I,J, Ȋ \J,A , (2.6) Π [s I ,r I ],I, J, Ȋ \J,A , (2.7) Π [s I ,r I ],I, J, Ȋ \J,A , (2.8) Π [s I ,r I ],I,J, I\J,A , (2.9) Π [s I ,r I ],I, J, I\J,A , (2.10) Π [s I ,r I ],I, J, I\J,A .

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Drinfeld's Lemma for Perfectoid Spaces and Overconvergence of Multivariate $(\phi, \Gamma)$-Modules

Cited by 2 publications

References 17 publications

Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

Analytic Geometry and Hodge-Frobenius Structure Continued

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