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?

## 1 Answer

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.