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The analytic de Rham stack in rigid geometry

Juan Esteban Rodríguez Camargo

Abstract

Applying the new theory of analytic stacks of Clausen and Scholze we introduce a general notion of derived Tate adic spaces. We use this formalism to define the analytic de Rham stack in rigid geometry, extending the theory of $D$-cap-modules of Ardakov and Wadsley to the theory of analytic $D$-modules. We prove some foundational results such as the existence of a six functor formalism and Poincaré duality for analytic $D$-modules, generalizing previous work of Bode. Finally, we relate the theory of analytic $D$-modules to previous work of the author with Rodrigues Jacinto on solid locally analytic representations of $p$-adic Lie groups.

The analytic de Rham stack in rigid geometry

Abstract

Applying the new theory of analytic stacks of Clausen and Scholze we introduce a general notion of derived Tate adic spaces. We use this formalism to define the analytic de Rham stack in rigid geometry, extending the theory of -cap-modules of Ardakov and Wadsley to the theory of analytic -modules. We prove some foundational results such as the existence of a six functor formalism and Poincaré duality for analytic -modules, generalizing previous work of Bode. Finally, we relate the theory of analytic -modules to previous work of the author with Rodrigues Jacinto on solid locally analytic representations of -adic Lie groups.
Paper Structure (51 sections, 163 theorems, 334 equations)

This paper contains 51 sections, 163 theorems, 334 equations.

Table of Contents

  1. Introduction
  2. Derived Tate adic spaces
  3. Preliminaries
  4. Analytic $\mathbb{E}_{\infty}$-rings
  5. Generalized Huber pairs
  6. Categorified locale
  7. Tate adic spaces as categorified locale
  8. Some idempotent algebras
  9. Condensed Nil-radical
  10. Bounded affinoid rings and $\dagger$-nil-radicals
  11. Adic spectrum and derived Tate adic spaces
  12. Derived Tate adic spaces
  13. Analytification functor
  14. Tate stacks
  15. Recollections on abstract six functor formalisms
  16. ...and 36 more sections

Key Result

Theorem 1.0.1

Let $R=\mathbb{Z}((\pi))$ be the Huber ring parametrizing pseudo-uniformizers in Tate Huber pairs. There is a full subcategory $\mathop{\mathrm{AffRing}}\nolimits^{b}_{R}\subset \mathop{\mathrm{AnRing}}\nolimits_R$ of the $\infty$-category of analytic $R$-algebras, called the category of bounded aff

Theorems & Definitions (430)

  • Theorem 1.0.1
  • Definition 1.0.2
  • Theorem 1.0.3
  • Theorem 1.0.4
  • Proposition 1.0.5
  • Definition 1.0.6
  • Theorem 1.0.7
  • Theorem 1.0.8
  • Corollary 1.0.9
  • Definition 1.0.10: Algebraic de Rham stack
  • ...and 420 more