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

Abstract

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 $\text{Fdiv}$ of $F$–divided objects. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$.