Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local structure of smooth \texorpdfstring{$p$}{p}-adic analytic Artin stacks

Amos Kaminski

Abstract

We prove \cite[Conjecture~5.17]{Clausen} on the local light--profinite structure of smooth $p$-adic analytic Artin stacks. The argument proceeds in several reductions. First, by proving a generalization of van~Dantzig theorem for groupoids, we reduce the conjecture to the compact Hausdorff case. This reduces the conjecture to the statement that the geometric realization of a groupoid object whose object and morphism spaces are light profinite and whose source and target maps are open is light profinite. Next, we simplify the groupoid by constructing a closed skeleton; after quotienting by a clopen subgroupoid, the remaining problem reduces to proving that a profinite family of finite groups can be presented as an inverse limit of finite families of finite groups. As observed by Clausen immediately after \cite[Conjecture~5.17]{Clausen}, our result implies in particular that smooth $p$-adic analytic Artin stacks are \emph{$!$-good}.

Local structure of smooth \texorpdfstring{$p$}{p}-adic analytic Artin stacks

Abstract

We prove \cite[Conjecture~5.17]{Clausen} on the local light--profinite structure of smooth -adic analytic Artin stacks. The argument proceeds in several reductions. First, by proving a generalization of van~Dantzig theorem for groupoids, we reduce the conjecture to the compact Hausdorff case. This reduces the conjecture to the statement that the geometric realization of a groupoid object whose object and morphism spaces are light profinite and whose source and target maps are open is light profinite. Next, we simplify the groupoid by constructing a closed skeleton; after quotienting by a clopen subgroupoid, the remaining problem reduces to proving that a profinite family of finite groups can be presented as an inverse limit of finite families of finite groups. As observed by Clausen immediately after \cite[Conjecture~5.17]{Clausen}, our result implies in particular that smooth -adic analytic Artin stacks are \emph{-good}.
Paper Structure (12 sections, 28 theorems, 79 equations, 1 figure)

This paper contains 12 sections, 28 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Light condensed anima
  4. p-adic analytic smooth Artin stacks
  5. Étale locality
  6. Preparatory Theorems
  7. Characterization of light profinite anima
  8. Global sections detect equivalences of coherent 1-pyknotic anima
  9. F-analytic Lie Groupoid
  10. Profinite Groupoids and Van Dantzig Theorem
  11. Profiniteness
  12. Proof of Conjecture 5.17

Key Result

Theorem A

Let $X$ be a smooth $p$-adic analytic Artin stack. Then étale-locally the associated light condensed anima $L(X)$ is a light profinite anima.

Figures (1)

  • Figure 1: Claude Monet - The Gare Saint-Lazare, Arrival of a Train

Theorems & Definitions (60)

  • Theorem A: Clausen -- Theorem \ref{['thm:clausen-conj']} in the text
  • Theorem B: Theorem \ref{['thm:Gamma-conservative-1type']} in the text
  • Proposition C: Proposition \ref{['light']} in the text
  • Theorem D: Theorem \ref{['VanDan']} in the text
  • Lemma E: Lemma \ref{['lem:skeletal-replacement']} in the text
  • definition 1: Clausen
  • definition 2: Clausen
  • Remark
  • definition 3: Clausen
  • Proposition 4: Clausen
  • ...and 50 more