# Frobenius and regular scheme

Let $$X$$ be a noetherian regular scheme over $$\mathbb{F}_{p}$$. Then by Kunz's theorem, the absolute Frobenius $$F: X\rightarrow X$$ is flat and integral. Can it be written as a projective limit of finite flat morphisms?

This is true Zariski-locally by Popescu's desingularisation theorem [Tag 07GC]. Indeed, any regular $$\mathbf F_p$$-algebra $$A$$ is geometrically regular [Tag 0381], so by Popescu's theorem it can be written as a filtered colimit $$\underset \to{\operatorname{colim}} B_i$$ of smooth $$\mathbf F_p$$-algebras $$B_i$$. Writing $$X = \operatorname{Spec} A$$ and $$Y_i = \operatorname{Spec} B_i$$, we get $$X \cong \lim_{\substack{\longleftarrow \\ i}} Y_i.$$ For each map $$X \to Y_i$$, consider the commutative diagram $$\begin{array}{ccccc}X & \!\!\overset{F_{X/Y_i}}\to\!\! & X_i & \!\!\overset{F_i}\to\!\! & X \\ & \!\!\searrow\!\! & \downarrow & & \downarrow \\ & & Y_i & \!\!\underset {F_{Y_i}}\to\!\! & Y_i\end{array}\label{dia frob}\tag{1}$$ where $$F_{Y_i} \colon Y_i \to Y_i$$ is the absolute Frobenius, the right square is a pullback, and the composition of the top maps is the absolute Frobenius $$F_X \colon X \to X$$. Note that $$F_{Y_i}$$ is finite flat since $$Y_i \to \operatorname{Spec} \mathbf F_p$$ is smooth, hence $$F_i$$ is finite flat as well. Taking limits of \eqref{dia frob} over all $$i$$ gives a commutative diagram $$\begin{array}{ccccc}X & \!\!\to\!\! & \displaystyle\lim_{\substack{\longleftarrow \\ i}}X_i & \!\!\to\!\! & X \\ & \!\!\searrow\!\! & \downarrow & & \downarrow \\ & & X & \!\!\underset {F_X}\to\!\! & X\end{array}$$ where the left diagonal and the right vertical arrows are isomorphisms, the right square is a pullback, and the composition of the top maps is $$F_X$$. We conclude that the middle vertical arrow is an isomorphism as well (since the diagram is a pullback), hence so is $$X \to \lim_i X_i$$ (2-out-of-3), identifying $$F_X \colon X \to X$$ with the limit of the finite flat maps $$X_i \to X$$. $$\square$$
If you want to globalise this, one method would be to produce a global Néron–Popescu desingularisation (for a regular Noetherian $$\mathbf F_p$$-scheme $$X$$). I don't know if this has been studied (nor can I construct this myself on the spot). Since any Noetherian $$\mathbf F_p$$-scheme is a limit of finite type $$\mathbf F_p$$-schemes and the desingularisation argument of Popescu goes by Noetherian induction, it is not completely inconceivable that one might be able to do this. But the construction involves a bunch of choices and computations, and it is not at all clear to me how to globalise each step.