On Non-Commutative Analytic Spaces Over Non-Archimedean Fields

Soibelman, Yan

doi:10.1007/978-3-540-68030-7_7

Cited by 11 publications

(22 citation statements)

References 11 publications

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“…Now we assume A to be some noncommutative Banach affinoid algebra over the local fields we consider above. Examples of such rings could be coming from the corresponding context of [Soi1], or the corresponding noncommutative Fréchet-Stein algebras as in [ST1]. We will use the notation E{Z 1 , ..., Z n } to denote the corresponding noncommutative non-noetherian Tate algebra as in the commutative setting.…”

Section: [Sr]mentioning

confidence: 99%

Hodge-Iwasawa Theory II

Tong

2020

Preprint

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We continue our study on the Hodge-Iwasawa theory which is a continuation of our previous work on Hodge-Iwasawa theory, which is aimed at higher dimensional deformation of higher dimensional Hodge structures over general analytic spaces or adic spaces.We still follow closely the approaches of Kedlaya-Liu to study our Frobenius modules over the different kinds of period rings including more generalized perfect period rings and the corresponding imperfect period rings. It is desirable that one can globalize the deformed information in order to organize the deformed period rings into the corresponding sheaves with respect to the coefficient sheaves, which will build up some foundations on further application to moduli of Frobenius sheaves.

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Section: [Sr]mentioning

confidence: 99%

Hodge-Iwasawa Theory II

Tong

2020

Preprint

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“…This could fit in well with the algebraic analysis of Kashiwara, Schapira, Saito and others. Non-commutative versions may be possible as well along the lines of [45,28]. It would also be nice to be able to work over rings instead of fields, for instance the rings of p-adic intgers Z p and the integers equipped with either the discrete or the standard norm and do types of analytic geometry over these rings.…”

Section: Work In Progressmentioning

confidence: 99%

“…Two of the theorems which we prove (Theorems 5. 16 and 5.31) show that that the finitely presented morphisms of affinoid algebras A → B which are homotopy epimorphisms (meaning that B ⊗ [45]. He identifies this question as being a key step in the development of noncommutative analytic geometry.…”

Section: Introductionmentioning

confidence: 99%

Non-Archimedean analytic geometry as relative algebraic geometry

Ben-Bassat¹,

Kremnizer²

2017

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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We show that non-Archimedean analytic geometry can be viewed as relative algebraic geometry in the sense of Toën-Vaquié-Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders. ContentsOREN BEN-BASSAT, KOBI KREMNIZER A.5. Enough projectives and injectives 47 A.6. The closed structure in the category of Banach spaces 51 A.7. Completion 51 A.8. Banach algebras and modules 52 Appendix B. Category theory background 54 References 55

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“…We note in passing that Soibelman's quantum affinoid algebras K{T } q,r appearing in [75] and [76] are examples of almost commutative affinoid K-algebras not of the form A n,K for some almost commutative R-algebra A, provided that |q −1| < 1.…”

Section: Almost Commutative Affinoid Algebrasmentioning

confidence: 99%

On irreducible representations of compact p-adic analytic groups

2013

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We prove that the canonical dimension of a coadmissible representation of a semisimple p-adic Lie group in a p-adic Banach space is either zero or at least half the dimension of a non-zero coadjoint orbit. To do this we establish analogues for p-adically completed enveloping algebras of Bernstein's inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen's Lemma about the endomorphism ring of a simple module over an enveloping algebra.

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On Non-Commutative Analytic Spaces Over Non-Archimedean Fields

Cited by 11 publications

References 11 publications

Hodge-Iwasawa Theory II

Hodge-Iwasawa Theory II

Non-Archimedean analytic geometry as relative algebraic geometry

On irreducible representations of compact p-adic analytic groups

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