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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

Let 𝒳S be a flat algebraic stack of finite presentation. We define a new étale fundamental pro-groupoid Π1(𝒳S), 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 Π1(𝒳S) naturally arises as colimit of the system of relative Frobenius morphisms 𝒳 𝒳pS 𝒳p2S in the pro-category of Deligne Mumford stacks. We give an interpretation of this result as an adjunction between Π1 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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relative Frobenius, $F$–divided object, perfection, coperfection, étale fundamental group, étale affine hull
Mathematical Subject Classification
Primary: 13A35, 14D23, 14F35, 14G17
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
Yuliang Huang
Chengdu Research Institute
Huawei Technology
Giulio Orecchia
École polytechnique fédérale de Lausanne
Matthieu Romagny
Université de Rennes 1