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

Abstract

Fix a scheme $S$ of characteristic $p$. Let $\mathcal{M}$ be an $S$-algebraic stack and let $\mbox{Fdiv}(\mathscr{M})$ be the stack of $\mbox{F}$-divided objects, that is sequences of objects $x_i\in \mathcal{M}$ with isomorphisms $σ_i:x_i\to \mbox{F}^*x_{i+1}$. Let $\mathcal{X}$ be a flat, finitely presented $S$-algebraic stack and $\mathcal{X}\to {\Pi }_1(\mathcal{X}/S)$ the étale fundamental pro-groupoid, constructed in the present text. We prove that if $\mathcal{M}$ is a quasi-separated Deligne-Mumford stack and $\mathcal{X}\to S$ has geometrically reduced fibres, there is a bifunctorial isomorphism of stacks \[\mathscr{H}\!om(Π_1(\mathscr{X}/S),\mathscr{M}) \simeq \mathscr{H}\!om(\mathscr{X},\mbox{Fdiv}(\mathscr{M})).\] In particular, the system of relative Frobenius morphisms $\mathcal{X}\to {\mathcal{X}}^{p/S}\to {\mathcal{X}}^{p^2/S}\to \dots$ allows to recover the space of connected components ${\pi }_0(\mathcal{X}/S)$ and the relative étale fundamental gerbe. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$.

50 pages. Improved the main result and fixed some proofs in section 5. Comments are welcome