Groupes analytiques rigides p-divisibles

Fargues, Laurent

doi:10.1007/s00208-018-1782-9

Cited by 12 publications

(31 citation statements)

References 12 publications

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“…We now form the fibre over the identity section Spa(K) → G. Since K is algebraically closed, this results by [43,Lemma 7.19] in a diagram S N/N ǫ Spa(K) in which the dotted arrow is a surjective map of pro-étale sheaves. This is a contradiction as we can see on (C, C + )-points: Any (C, C + )-point of S comes from a K-point, but since d := dim G > 0, the sheaf N/N ǫ has many points which do not come from K-points after any extension: For example, by [20,Proposition 9] this contains a subsheaf of the form B d δ /B d ǫ for some 1 > δ > ǫ, whose (C, C + )-points are (̟ δ C + /̟ ǫ C + ) d . Now use that d > 0.…”

Section: V-sheaves Associated To Schemes Over the Residue Fieldmentioning

confidence: 88%

Line bundles on perfectoid covers: case of good reduction

Heuer¹

2021

Preprint

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We study Picard groups in profinite perfectoid limits of rigid spaces in the case of good reduction. Our main result is that for sufficiently large covers, these can be described in terms of the special fibre: For example, our description implies that for a supersingular variety X of good reduction X over a perfectoid field of characteristic p,Via Raynaud uniformisation, it moreover allows us to describe Picard groups of proétale covers of abeloid varieties, which has applications to pro-étale uniformisation. We use our results to answer some general questions about Picard groups and Picard functors of perfectoid spaces, e.g. we show that these are not always p-divisible. Along the way we construct a "multiplicative Hodge-Tate spectral sequence" for O × . This is part III of a series on line bundles on diamonds.

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Section: V-sheaves Associated To Schemes Over the Residue Fieldmentioning

confidence: 88%

Line bundles on perfectoid covers: case of good reduction

Heuer¹

2021

Preprint

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“…In the rigid case, G p ∞ may often be identified with an analytic p-divisible subgroup in the sense of Fargues [Far19], as we shall now discuss. We first recall Fargues' perspective:…”

Section: Topologically Torsion Subgroups Of Rigid Groupsmentioning

confidence: 99%

Diamantine Picard functors of rigid spaces

Heuer¹

2021

Preprint

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For a connected smooth proper rigid space X over a perfectoid field extension of Qp, we show that the Picard functor of X ♦ defined on perfectoid test objects is the diamondification of the rigid analytic Picard functor. In particular, it is represented by a rigid group variety if and only if the rigid analytic Picard functor is.As an application, we determine which line bundles are trivialized by pro-finiteétale covers, and prove unconditionally that the associated "topological torsion Picard functor" is represented by a divisible analytic group. We use this to generalize and geometrize a construction of Deninger-Werner in the p-adic Simpson correspondence: There is an isomorphism of rigid analytic group varieties between the moduli space of continuous characters of π1(X, x) and that of pro-finite-étale Higgs line bundles on X.This article is part II of a series about line bundles on rigid spaces as diamonds.1. The natural map (Pic X,ét ) ♦ → Pic ♦ X,ét is an isomorphism of sheaves on Perf K,ét .2. If moreover K is algebraically closed, then the v-Picard functor Pic ♦ X,v fits into a split exact sequence of abelian sheaves on Perf K,ét 0 → Pic ♦ X,ét → Pic ♦ X,v → H 0 (X, Ω 1 )(−1) ⊗ K G a → 0.

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“…The theory of Banach-Colmez spaces was refined and extended by Plût [67], and, more recently, by Fargues-Fontaine [35], Fargues [34], and Le Bras [56]. Banach-Colmez spaces are a special case of Scholze's diamonds [72].…”

Section: Banach-colmez Spaces We Have Howevermentioning

confidence: 99%

Hodge theory of $p$-adic varieties: a survey

Nizioł¹

2021

Ann. Polon. Math.

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Groupes analytiques rigides p-divisibles

Cited by 12 publications

References 12 publications

Line bundles on perfectoid covers: case of good reduction

Line bundles on perfectoid covers: case of good reduction

Diamantine Picard functors of rigid spaces

Hodge theory of $p$-adic varieties: a survey

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