Analytification, localization and homotopy epimorphisms

Ben-Bassat, Oren; Mukherjee, Devarshi

doi:10.48550/arxiv.2111.04184

Cited by 2 publications

(5 citation statements)

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“…We now take the corresponding colimit through all the (∞, 1)-categories. Therefore all the corresponding (∞, 1)-functors into (∞, 1)categories or (∞, 1)-groupoids are from the homotopy closure of Q p C 1 , ..., C ℓ ℓ = 1, q, ... in sCommIndBanach Q p or Q p C 1 , ..., C ℓ ℓ = 1, q, ... in sCommInd m Banach Q p as in [BBM,Section 4.2]: Ind Q p C 1 ,...,C ℓ ,ℓ=1,q,... sCommIndBanach Q p , (3.2.5)…”

Section: ∞-Categorical Analytic Stacks and Descents IVmentioning

confidence: 99%

$\infty$-Categorical Approaches to Hodge-Iwasawa Theory I: Introduction and Extensions

Tong¹

2022

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In this paper, we give the introduction to the Hodge-Iwasawa Theory introduced by the author. After that we will give some well-defined extensions to the already shaped framework established in our previous work. Scholze's v-Spaces and Six FormalismScholze's v-stacks happen over the category of perfectoid space over F p . We give the introduction following closely [Sch3] and [SW]. Our presentation is also following closely [Sch3].Setting 1.1.1. v-sheaves and v-stacks carry the topology which is called the v-topology which is finer than analytic topology, étale topology, pro-étale topology.Example 1.1.2. The two important situations are the following. First is the situation where the v-stack carries a basis of topological neighbourhoods consisting of perfectoids. The second situation is the key moduli of vector bundles over FF curves in [FS]. Definition 1.1.3. (Scholze [Sch3], Analytic Prestacks) Consider the following Grothendieck sites: Perfectoid F p ,étale , Perfectoid F p ,proétale , Perfectoid F p ,v . (1.1.1) We just define a (2, 1)-presheaf F over these sites to be any functor from Perfectoid F p ,étale , Perfectoid F p ,proétale , Perfectoid F p ,v . (1.1.2) to the groupoids. Definition 1.1.4. (Scholze [Sch3], Analytic Stacks) Consider the following Grothendieck sites: Perfectoid F p ,étale , Perfectoid F p ,proétale , Perfectoid F p ,v . (1.1.3)We just define a (2, 1)-sheaf F over these sites to be any functor from Perfectoid F p ,étale , Perfectoid F p ,proétale , Perfectoid F p ,v .(1.1.4) to the groupoids, which is further a stack in groupoids.We then have the morphisms and sites of analytic stacks in certain situations. Definition 1.1.5. (Scholze [Sch3, Definition 1.20], Morphisms of Analytic Stacks) The étale and quasi-pro-étale morphisms between small v-stacks are defined by using perfectoid coverings, and defining the corresponding étaleness and quasi-pro-étaleness after taking base changes along such perfectoid coverings. Definition 1.1.6. (Scholze [Sch3, Definition 26.1], Sites of Analytic Stacks) Let X be a small v-stack, we have the v-site X v 2 . This then gives the ∞-category D I (X v , λ) of derived I-complete objects in ∞-category of all λ-sheaves D(X v , λ) where λ is a derived I-commutative algebra object. Theorem 1.1.7. (Scholze [Sch3, Definition 26.1, Before the Remark 26.3]) This ∞-category D I (X v , λ) as well as the corresponding sub ∞-categories D I,qsproét (X v , λ) and D I,ét (X v , λ) of the corresponding derived I-complete objects over the quasi-proétale and étale sites of small vstacks admit six formalism through: ⊗, Hom, f ! , f ! , f * , f * as in [Sch3, Definition 26.1, Before the Remark 26.3]. Then as in [FS] for the corresponding solid ∞-category we can say the parallel things: Theorem 1.1.8. (Fargues-Scholze [FS, Chapter VII.2]) This ∞-category D (X v , λ) admits four formalism as in [FS, Chapter VII.2]. Here the assumption on λ will be definitely weaker solid condensed ring as assumed in [FS, Beginning of Chapter VII.2].Remark 1.1.9. In our situation, certainly what we...

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Section: ∞-Categorical Analytic Stacks and Descents IVmentioning

confidence: 99%

$\infty$-Categorical Approaches to Hodge-Iwasawa Theory I: Introduction and Extensions

Tong¹

2022

Preprint

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“…The corresponding analogs of analytification functors from Ben-Bassat-Mukherjee [BBM,Section 4.2] are given in the following. First we consider the ∞-category of E 1 -rings from [Lu2,Proposition 7.1.4.18, as well as the discussion above Proposition 7.1.4.18 on page 1225] which we denote it by Noncommutative E 1 ,Simplicial , then we consider the corresponding category of all the polynomial rings with free variables over R, which we denote it by Polynomial free R , then we have the corresponding fully faithful embedding:…”

Section: Notations On ∞-Categories Of Noncommutative ∞-Ringed Toposesmentioning

confidence: 99%

“…As in [BBM,Section 4.2], this will give the process what we call smooth formal series analytification by taking into account the corresponding homotopy colimit completion:…”

Section: Notations On ∞-Categories Of Noncommutative ∞-Ringed Toposesmentioning

confidence: 99%

“…The resulting ∞-stacks generated are interesting to study. We call the corresponding functors are derived functional analytic Hochschild cohomology presheaves, derived functional analytic period cohomology presheaves and derived functional analytic cyclic cohomology presheaves, which we are going to denote these presheaves as in the following for 12 Definitely, we need to put certain norms over in some relatively canonical way, as in [BBM,Section 4.2] one can basically consider rigid ones and dagger ones, and so on. We restrict to the formal one.…”

Section: R Lim ← −mentioning

confidence: 99%

“…One has the corresponding p-completed versions as well.22 Definitely, we need to put certain norms over in some relatively canonical way, as in[BBM, Section 4.2] one can basically consider rigid ones and dagger ones, and so on. In this noncommutative setting we do not actually need to fix the type of the analytification, as in commutative setting since we are going to apply directly construction from Bhatt-Scholze and Koshikawa[BS1] and[Ko1], though the derived characterization of the prismatic cohomology might not need to be restricted to the corresponding p-adic formal scheme situations.23 In all the following, we assume this prism to be bounded and satisfy that A/I is Banach.…”

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Topologization and Functional Analytification II: $\infty$-Categorical Motivic Constructions for Homotopical Contexts

Tong¹

2021

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This is our second scope of the consideration on the corresponding topologization and the corresponding functional analytification. We will focus on the corresponding functorial and motivic constructions in our current consideration. We consider topological motivic derived I-adic and derived (p, I)-adic cohomologies through derived de Rham complexes of Bhatt, Guo, Illusie, Morrow, Scholze, Frobenius sheaves over Robba rings of Kedlaya-Liu in certain derived I-adic and derived (p, I)-adic geometric context as what we defined for Bambozzi-Ben-Bassat-Kremnizer ∞-prestacks in our previous work in this series. The foundation we will work on will be based on the work of Bambozzi-Ben-Bassat-Kremnizer, Ben-Bassat-Mukherjee, Clausen-Scholze and Kelly-Kremnizer-Mukherjee, in order to promote the construction to even more general homotopical and ∞-categorical contexts. This gives us the chance to construct the functional analytic derived prismatic cohomology and derived preperfectoidizations, as well as the functional analytic derived logarithmic prismatic cohomology and derived logarithmic preperfectoidizations after Bhatt-Scholze and Koshikawa, in the framework of Bambozzi-

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Analytification, localization and homotopy epimorphisms

Cited by 2 publications

References 13 publications

$\infty$-Categorical Approaches to Hodge-Iwasawa Theory I: Introduction and Extensions

$\infty$-Categorical Approaches to Hodge-Iwasawa Theory I: Introduction and Extensions

Topologization and Functional Analytification II: $\infty$-Categorical Motivic Constructions for Homotopical Contexts

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