Analytic geometry over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> and the Fargues-Fontaine curve

Bambozzi, Federico; Ben-Bassat, Oren; Kremnizer, Kobi

doi:10.1016/j.aim.2019.106815

Cited by 12 publications

(35 citation statements)

References 19 publications

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“…Also we have the following foundations. First we have from [BBBK,Proposition 1.6] the Banach completion on the Banach sets over some Banach ring R which is the left adjoint functor of the inclusion:…”

Section: ∞-Categorical Completion and ∞-Categorical Solidificationmentioning

confidence: 99%

$\infty$-Categorical Functional Analysis and $p$-adic Motives

Tong¹

2021

Preprint

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We discuss the deep relationship between ∞-categorical functional analysis and the anticipated theory of p-adic motives. The motivation fundamentally comes from applications essentially in arithmetics from very broad perspectives. The ∞-categorical homotopical aspects we considered here come mainly from Bambozzi-Ben-Bassat-Kremnizer, Ben-Bassat-Mukherjee, Bambozzi-Kremnizer, Clausen-Scholze and Kelly-Kremnizer-Mukherjee, based on deep robust foundation of ∞-categorical solids and ∞-categorical ind-Banach or bornological modules over general Banach rings.

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Section: ∞-Categorical Completion and ∞-Categorical Solidificationmentioning

confidence: 99%

$\infty$-Categorical Functional Analysis and $p$-adic Motives

Tong¹

2021

Preprint

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“…We recall that a normed set is a set S equipped with a map | • | S : S → [0, ∞), and NSet denotes the category of normed sets and bounded maps. See [BBK19] §1 for more details about this category. For an S ∈ NSet, we put S + ≔ {s ∈ S | s S > 0}.…”

Section: Homological Algebra For Discrete Banach Modulesmentioning

confidence: 99%

“…The richness of this theory comes from the structure of C as a complete valued field with respect to the standard absolute value. In the authors' recent work [BM21], it has been shown that the theory of C * -algebras has a nice description in terms of the derived analytic geometry introduced in [BK17], [BB16], [BBK19] and [BK20]. Therefore, it is natural to ask what happens when the continuous C-valued functions on X are replaced by the Banach algebra C(X, R) of continuous functions valued on other Banach rings.…”

Section: Introduction Backgroundmentioning

confidence: 99%

Derived Analytic Geometry for Z-Valued Functions. Part I -- Topological Properties

Bambozzi¹,

Mihara²

2021

Preprint

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We study the Banach algebras C(X, R) of continuous functions from a compact Hausdorff topological space X to a Banach ring R whose topology is discrete. We prove that the Berkovich spectrum of C(X, R) is homeomorphic to ζ(X) × M (R), where ζ(X) is the Banaschewski compactification of X and M (R) is the Berkovich spectrum of R. We study how the topology of the spectrum of C(X, R) is related to the notion of homotopy Zariski open embedding used in derived geometry. We find that the topology of ζ(X) can be easily reconstructed from the homotopy Zariski topology associated to C(X, R). We also prove some results about the existence of Schauder bases on C(X, R) and a generalisation of the Stone-Weierstrass Theorem, under suitable hypotheses on X and R.

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“…In this section, we recall the basic properties of the quasi-Abelian category of Banach modules and recall the foundation of the derived analytic geometry as discussed in [BB16], [BK17], [BBK19], and [BK20].…”

Section: Preliminariesmentioning

confidence: 99%

“…This approach has been recently developed in the series of papers (cf. [BB16], [BK17], [BBK19], and [BK20]). Once the notions of derived categories and derived functors are correctly extended to the analytic setting, it is natural to consider the derived variant of Tate's acyclicity.…”

Section: Introductionmentioning

confidence: 99%

Homotopy Epimorphisms and Derived Tate's Acyclicity for Commutative C*-algebras

Bambozzi¹,

Mihara²

2021

Preprint

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We study homotopy epimorphisms and covers formulated in terms of derived Tate's acyclicity for commutative C * -algebras and their non-Archimedean counterparts. We prove that a homotopy epimorphism between commutative C * -algebras precisely corresponds to a closed immersion between the compact Hausdorff topological spaces associated to them, and a cover of a commutative C * -algebra precisely corresponds to a topological cover of the compact Hausdorff topological space associated to it by closed immersions admitting a finite subcover. This permits us to prove derived and non-derived descent for Banach modules over commutative C * -algebras.

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Analytic geometry over F1 and the Fargues-Fontaine curve

Cited by 12 publications

References 19 publications

$\infty$-Categorical Functional Analysis and $p$-adic Motives

$\infty$-Categorical Functional Analysis and $p$-adic Motives

Derived Analytic Geometry for Z-Valued Functions. Part I -- Topological Properties

Homotopy Epimorphisms and Derived Tate's Acyclicity for Commutative C*-algebras

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