Unramified F–divided objects and the étale fundamental pro-groupoid in positive characteristic

Huang, Yuliang; Orecchia, Giulio; Romagny, Matthieu

doi:10.2140/gt.2022.26.3221

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Unramified $F$–divided objects and the étale fundamental pro-groupoid in positive characteristic

Yuliang Huang, Giulio Orecchia and Matthieu Romagny

Geometry & Topology 26 (2022) 3221–3306

DOI: 10.2140/gt.2022.26.3221

Abstract

Let $𝒳 ∕ S$ be a flat algebraic stack of finite presentation. We define a new étale fundamental pro-groupoid $Π_{1} (𝒳 ∕ S)$ , generalizing Grothendieck’s enlarged étale fundamental group from SGA 3 to the relative situation. When $S$ is of equal positive characteristic $p$ , we prove that $Π_{1} (𝒳 ∕ S)$ naturally arises as colimit of the system of relative Frobenius morphisms $𝒳 \to 𝒳^{p ∕ S} \to 𝒳^{p^{2} ∕ S} \to \dots$ in the pro-category of Deligne Mumford stacks. We give an interpretation of this result as an adjunction between $Π_{1}$ and the stack $Fdiv$ of $F$ –divided objects. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$ .

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Keywords

relative Frobenius, $F$–divided object, perfection, coperfection, étale fundamental group, étale affine hull

Mathematical Subject Classification

Primary: 13A35, 14D23, 14F35, 14G17

References

Publication

Received: 19 February 2021

Revised: 6 August 2021

Accepted: 4 September 2021

Published: 23 January 2023

Proposed: Dan Abramovich

Seconded: Richard P Thomas, Marc Levine

Authors

Yuliang Huang
Chengdu Research Institute Huawei Technology Chengdu China

Giulio Orecchia
École polytechnique fédérale de Lausanne Lausanne Switzerland

Matthieu Romagny
CNRS, IRMAR – UMR 6625 Université de Rennes 1 Rennes France

Volume 26, issue 7 (2022)