Towards a Non-Archimedean Analytic Analog of the Bass–quillen Conjecture

Kerz, Moritz; Saito, Shuji; Tamme, Georg

doi:10.1017/s147474801900001x

Cited by 6 publications

(5 citation statements)

References 14 publications

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“…As a corollary to Proposition 5.14 and Proposition 5.10(ii) we recover the following pro-homotopy invariance result for K 0 , which has already been shown in [KST19] using a more direct approach.…”

Section: Analytic Pro-homotopy Invariancesupporting

confidence: 82%

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$K$-Theory of Non-Archimedean Rings. I

2019

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We introduce a variant of homotopy K-theory for Tate rings, which we call analytic K-theory. It is homotopy invariant with respect to the analytic affine line viewed as an ind-object of closed disks of increasing radii. Under a certain regularity assumption we prove an analytic analog of the Bass fundamental theorem and we compare analytic K-theory with continuous K-theory, which is defined in terms of models. Along the way we also prove some results about the algebraic K-theory of Tate rings.

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Section: Analytic Pro-homotopy Invariancesupporting

confidence: 82%

“…So in some sense this new homotopy theory keeps more information than the previous approaches mentioned above. Our approach is motivated by the observation that the Picard group of a smooth affinoid algebra is homotopy invariant in our sense, while it is not B κ -invariant in general, see [KST19].…”

Section: Analytic K-theory and Karoubi-villamayor K-theorymentioning

confidence: 99%

$K$-Theory of Non-Archimedean Rings. I

2019

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“…a) Every line bundle over X × A 1 rig has a local trivialisation by subsets of the form {U i × A 1 rig } i∈J where {U i } i∈J is an admissible covering of X. This is Corollary 3.9, a corollary of Theorem 3.7 by Kerz, Saito and Tamme [KST16]. b) Consider a line bundle on X × A 1 rig and a local trivialisation of the form {U i × A 1 rig } i∈J .…”

Section: Introductionmentioning

confidence: 90%

Homotopy Classification of Line Bundles Over Rigid Analytic Varieties

Sigloch

2017

Preprint

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We construct a motivic homotopy theory for rigid analytic varieties with the rigid analytic affine line A 1 rig as an interval object. This motivic homotopy theory is inspired from, but not equal to, Ayoub's motivic homotopy theory for rigid analytic varieties. Working in the so constructed homotopy theory, we prove that a homotopy classification of vector bundles of rank n over rigid analytic quasi-Stein spaces follows from A 1 rig -homotopy invariance of vector bundles. This A 1 rig -homotopy invariance is equivalent to a rigid analytic version of Lindel's solution to the Bass-Quillen conjecture. Moreover, we establish a homotopy classification of line bundles over rigid analytic quasi-Stein spaces. In fact, line bundles are classified by infinite projective space.

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“…Néanmoins il se peut à priori que tout les S-groupes analytiques rigides p-divisibles ne proviennent pas de cette construction. En tentant d'adapter les arguments de la section 3 de [10] on tombe sur le problème de savoir si tout élément de Pic(B 1 S ) est localement trivial sur S, ce qui n'est pas connu même pour S lisse ( [17]).…”

Section: Familles De Groupes Analytiques Rigides P-divisiblesunclassified

Groupes analytiques rigides p-divisibles II

Fargues¹

2019

Preprint

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LAURENT FARGUESRÉSUMÉ. Soit K un corps p-adique. On continue de développer la théorie des groupes analytiques rigides p-divisibles sur K. On explique par exemple comment retrouver la catégorie des espaces de Banach-Colmez à partir des groupes analytiques rigides p-divisibles « en niveau fini » sans espaces perfectoïdes. On établit ensuite des résultats sur les familles de groupes analytiques rigides p-divisibles. Cela nous permet de démontrer un théorème de « minimalité » au sens de la géométrie birationnelle des modèles entiers des espaces de Rapoport-Zink non-ramifiés.ABSTRACT. Let K be a p-adic field. We continue to develop the theory of rigid analytic p-divisible groups over K. For example, we explain how to find back the category of Banach-Colmez spaces from rigid analytic p-divisible groups "in finite level" without perfectoid spaces. We then establish some results about families of rigid analytic p-divisible groups. This allows us to prove a "minimality" result in the sense of birationnal geometry for integral models of unramified Rapoport-Zink spaces.

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Towards a Non-Archimedean Analytic Analog of the Bass–quillen Conjecture

Cited by 6 publications

References 14 publications

$K$-Theory of Non-Archimedean Rings. I

$K$-Theory of Non-Archimedean Rings. I

Homotopy Classification of Line Bundles Over Rigid Analytic Varieties

Groupes analytiques rigides p-divisibles II

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