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Towards $\mathbb{A}^1$-homotopy theory of rigid analytic spaces

Christian Dahlhausen, Can Yaylali

TL;DR

The paper develops an $\mathbb{A}^1$-homotopy framework for rigid analytic spaces, constructing unstable and stable motivic categories with coefficients and establishing functorial and gluing properties that parallel the algebraic setting. It then proves a six-functor formalism for the stable theory after inverting Thom motives and, under a dagger-type assumption, shows representability results for analytic K-theory, including a Bass delooping and a $\mathbb{P}^1$-spectrum $K^{an}$ representing analytic K-theory. The analytification functor maps algebraic K-theory to analytic K-theory, and the theory identifies the connective part of analytic K-theory with $\mathbb{Z} \times \mathrm{BGL}$ in the unstable category, yielding a concrete representability picture. Finally, the authors develop continuous homotopy K-theory, prove it is the $\mathbb{A}^1$-localisation of continuous K-theory, and connect these constructions to pro- and condensed-spectra frameworks, offering a rich bridge between rigid analytic geometry and motivic homotopy theory.

Abstract

To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an $\mathbb{A}^1$-invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a functor on the category of rigid analytic spaces. Moreover, we show that there exists a full six functor formalism for the precomposition with the analytification functor by evoking Ayoub's thesis. As an application, we identify connective analytic K-theory in the unstable homotopy category with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of connective algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra.

Towards $\mathbb{A}^1$-homotopy theory of rigid analytic spaces

TL;DR

The paper develops an -homotopy framework for rigid analytic spaces, constructing unstable and stable motivic categories with coefficients and establishing functorial and gluing properties that parallel the algebraic setting. It then proves a six-functor formalism for the stable theory after inverting Thom motives and, under a dagger-type assumption, shows representability results for analytic K-theory, including a Bass delooping and a -spectrum representing analytic K-theory. The analytification functor maps algebraic K-theory to analytic K-theory, and the theory identifies the connective part of analytic K-theory with in the unstable category, yielding a concrete representability picture. Finally, the authors develop continuous homotopy K-theory, prove it is the -localisation of continuous K-theory, and connect these constructions to pro- and condensed-spectra frameworks, offering a rich bridge between rigid analytic geometry and motivic homotopy theory.

Abstract

To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an -invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a functor on the category of rigid analytic spaces. Moreover, we show that there exists a full six functor formalism for the precomposition with the analytification functor by evoking Ayoub's thesis. As an application, we identify connective analytic K-theory in the unstable homotopy category with both and the analytification of connective algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra.
Paper Structure (24 sections, 59 theorems, 130 equations)

This paper contains 24 sections, 59 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Background in rigid analytic geometry
  3. Unstable -homotopy theory in rigid geometry
  4. Homotopy invariant presheaves on rigid spaces
  5. The unstable analytic motivic category
  6. Analytification for motivic spaces
  7. Functoriality
  8. Gluing
  9. Stable -homotopy theory in rigid geometry
  10. Change of coefficients
  11. Effective motives and localisation
  12. Thom motives and stabilisation
  13. The relation with $\mathbb{P}^1$-spectra
  14. Applications to the K-theory of rigid spaces
  15. Continuous K-theory and analytic K-theory
  16. ...and 9 more sections

Key Result

Theorem 1

The assignment $X\mapsto \textup{Sh}^{\mathbb{A}^{1}}_{\textup{Nis}}(X,{\mathcal{V}}),\ f\mapsto f^{*}$, from rigid analytic varieties to symmetric monoidal presentable ${\mathcal{V}}$-linear categories satisfies (PF) and (Loc).

Theorems & Definitions (146)

  • Theorem 1: \ref{['prop-smooth-BC-and-PF']}, \ref{['thm-gluing']}
  • Corollary 2: \ref{['thm-6ff']}
  • Theorem 3: \ref{['thm:representability']}
  • Theorem 4: \ref{['bass-fundamental-theorem-for-Kan--cor']}
  • Corollary 5: \ref{['cor-K-P1-spectrum']}
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • ...and 136 more