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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
##### Abstract

Let $\mathsc{𝒳}∕S$ be a flat algebraic stack of finite presentation. We define a new étale fundamental pro-groupoid ${\mathrm{\Pi }}_{1}\left(\mathsc{𝒳}∕S\right)$, 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 ${\mathrm{\Pi }}_{1}\left(\mathsc{𝒳}∕S\right)$ naturally arises as colimit of the system of relative Frobenius morphisms $\mathsc{𝒳}\to {\mathsc{𝒳}}^{p∕S}\to {\mathsc{𝒳}}^{{p}^{2}∕S}\to \cdots$ in the pro-category of Deligne Mumford stacks. We give an interpretation of this result as an adjunction between ${\mathrm{\Pi }}_{1}$ and the stack $\mathrm{Fdiv}$ of $F$–divided objects. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$.

##### Keywords
relative Frobenius, $F$–divided object, perfection, coperfection, étale fundamental group, étale affine hull
##### Mathematical Subject Classification
Primary: 13A35, 14D23, 14F35, 14G17