Analytic Hochschild-Kostant-Rosenberg Theorem

Kelly, Jack B.; Kremnizer, Kobi; Mukherjee, Devarshi

doi:10.48550/arxiv.2111.03502

Cited by 3 publications

(4 citation statements)

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“…These are HA contexts by work in [9]. The more classical in [14], together with the fact (see Remark 7.2 )that L A ⊗ L A A an ∼ = L an A we have the following very general computation…”

Section: Hochschild Homologymentioning

confidence: 92%

Analytification, localization and homotopy epimorphisms

Ben-Bassat,

Mukherjee

2021

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We study the interaction between various analytification functors, and a class of morphisms of rings, called homotopy epimorphisms. An analytification functor assigns to a simplicial commutative algebra over a ring R, along with a choice of Banach structure on R, a commutative monoid in the monoidal model category of simplicial ind-Banach R-modules. We show that several analytifications relevant to analytic geometry -such as Tate, overconvergent, Stein analytification, and formal completion -are homotopy epimorphisms. Another class of examples of homotopy epimorphisms arises from Weierstrass, Laurent and rational localizations in derived analytic geometry. As applications of this result, we prove that Hochschild homology and the cotangent complex are computable for analytic rings, and the computation relies only on known computations of Hochschild homology for polynomial rings. We show that in various senses, Hochschild homology as we define it commutes with localizations, analytifications and completions.

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“…These are HA contexts by work in [9]. The more classical in [14], together with the fact (see Remark 7.2 )that L A ⊗ L A A an ∼ = L an A we have the following very general computation…”

Section: Hochschild Homologymentioning

confidence: 92%

Analytification, localization and homotopy epimorphisms

Ben-Bassat,

Mukherjee

2021

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“…In the following the right had of each row in each diagram will be the corresponding quasicoherent Robba bundles over the Robba ring carrying the corresponding action from the Frobenius or the fundamental groups, defined by directly applying [KKM,Section 9.3] and [BBM]. We now let A be any commutative algebra objects in the corresponding ∞-toposes over ind-Banach commutative algebra objects over Q p or the corresponding borné commutative algebra objects over Q p carrying the Grothendieck topology defined by essentially the corresponding monomorphism homotopy in the opposite category.…”

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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“…We would like to start from the corresponding context of [KKM, Section 5.2.1, Definition 5.5, Proposition 5.6, Definition 5.8, Definition 5.9] 2 , and represent the construction for the convenience of the readers 3 . Namely as in [KKM, Section 5.2.1, Definition 5.5, Proposition 5.6] we consider the corresponding cotangent complex associated to any pair object (A, B) ∈ X × X A is defined to be (as in [KKM,Section 5. Then we need to take the corresponding Hodge-Filtered completion by using the corresponding filtration associated as above:…”

Section: Chaptermentioning

confidence: 99%

“…The ∞-presheaves in this section in the ∞-category: are expected to be ∞-sheaves as long as one considers in the admissible situations the corresponding Čech ∞-descent for general seminormed modules as in [KKM,Section 9.3] and [BBK] such as Bambozzi-Kremnizer spaces in [BK]. Therefore we have: which is defined by taking the left Kan extension to all the (∞, 1)-ring objects in the ∞-derived category of all A-modules from formal series rings over A.…”

Section: R Lim ← −mentioning

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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Analytic Hochschild-Kostant-Rosenberg Theorem

Cited by 3 publications

References 23 publications

Analytification, localization and homotopy epimorphisms

Analytification, localization and homotopy epimorphisms

$\infty$-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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