Cohomology of $(φ,Γ)$-modules over pseudorigid spaces

Bellovin, Rebecca

doi:10.48550/arxiv.2102.04820

Cited by 3 publications

(4 citation statements)

References 15 publications

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“…• 2(∞-Categorical Rings and ∞-Categorical Spaces). The pseudorigid analytic affinoids and spaces over Z p are also interesting to study, which is a geometric version of the arithmetic family in [Bel1] and [Bel2];…”

Section: The Results Of [Xt]mentioning

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: The Results Of [Xt]mentioning

confidence: 99%

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

Tong¹

2022

Preprint

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“…Example 1.2.9. The second example is the corresponding pseudorigid analytic affinoids as in [Bel1,Definition 3.1], [Bel2] and [L,Definition 4.1]. They have open subrings which are actually derived t-adically complete for some specific pseudouniformizer t. Then one adds the corresponding prelog structures from [DLLZ1,Chapter 2].…”

Section: The Rings In Algebraic Topologymentioning

confidence: 99%

“…So one definitely has to be very careful in the corresponding analysis 24 . These kinds of spaces are actually different from the rigid analytic space, although the theories are really related to each other, such as in [Bel1], [Bel2] and [L]. Following the ideas in the rigid situation in [KL2], we consider the towers in the smooth situation in the following:…”

Section: Pseudorigid Relative Toric Towermentioning

confidence: 99%

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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“…• <3-> Here the product happens between the Kedlaya-Liu Robba ring with A is the Banach completed tensor product, one may want to consider the theory by taking the product ⊗ LSolidified . This may be very crucial when we want to take the product with any pseudorigid analytic affinoid as in Bellovin's work to treat some general eigenvarieties [Bel1], [Bel2]. Therefore this will be helpful when we want to take the product of rings having different topologies.…”

Section: Second Scope Of the Study: ∞-Categorical Analytic Hodge-iwas...mentioning

confidence: 99%

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

Tong¹

2021

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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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Cohomology of $(φ,Γ)$-modules over pseudorigid spaces

Cited by 3 publications

References 15 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

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

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