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

Abstract

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...