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

Huang, Yuliang¹ ; Orecchia, Giulio² ; Romagny, Matthieu³

¹ Chengdu Research Institute, Huawei Technology, Chengdu, China
² École polytechnique fédérale de Lausanne, Lausanne, Switzerland
³ CNRS, IRMAR – UMR 6625, Université de Rennes 1, Rennes, France

Geometry & topology, Tome 26 (2022) no. 7, pp. 3221-3306.

Résumé

Let $\mathcal{𝒳}∕S$ be a flat algebraic stack of finite presentation. We define a new étale fundamental pro-groupoid ${\mathrm{\Pi }}_1(\mathcal{𝒳}∕S)$, 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 ${\mathrm{\Pi }}_1(\mathcal{𝒳}∕S)$ naturally arises as colimit of the system of relative Frobenius morphisms $\mathcal{𝒳}\to {\mathcal{𝒳}}^{p∕S}\to {\mathcal{𝒳}}^{p^2∕S}\to \cdots$ 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 $p$.

Détail

DOI : 10.2140/gt.2022.26.3221

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

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