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

TL;DR

This paper proves a conjecture about the local structure of smooth p-adic analytic Artin stacks, showing they have a light-profinite structure through a series of reductions and groupoid analysis.

Contribution

It generalizes van Dantzig's theorem for groupoids and reduces the conjecture to a problem about inverse limits of finite groups, establishing the light-profinite structure.

Findings

Proved the local light-profinite structure of smooth p-adic analytic Artin stacks.

Reduced the conjecture to the case of compact Hausdorff groupoids.

Showed that such stacks are -good".

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}.