Let $X$ be a algebraic variety of finite type over $\mathbb{Z}$. Let $\mathcal{F}$ be a foliation in codimension one over $X$. Let $X_p$ and $\mathcal{F}_p$ be the reductions modulo $p$ of $X$ and $\mathcal{F}$ respectivelly. We suppose that for every affine open $U$ of $X_p$ and for every derivation $D$ in $\mathcal{F_p}(U)$ there exist a nontrivial solution of $D(x)=0$. Let $Y_p$ be the variety such that for every open $U$ in $X_p$ we have $Y_p(U)=\{ x\in \mathcal{O}_{X_p}(U)/ D(x)=0 \forall D \in \mathcal{F}(U)\}$.

Since $X_p$ is a totally ramify extension of $Y_p$, then we have the inclusion $i:X_p\mapsto Y_p$, the relative frobenius morphism $Fr_{Y_p}: Y_p\mapsto X_p$ and the absolute frobenius automorphism $Fr: X_p\mapsto X_p$. Let $\phi_p=i\circ Fr_{Y_p} \circ Fr^{-1}: X_p\mapsto X_p$. We observe that for an affine open $U$, $x$ is local solution of the problem $D(x)=0$ for every derivation $D\in \mathcal{F}(U)$ if and only if $\phi_p(U)(x)=x$.

I want to construct global solutions for the problem $D(x)=0$ for derivations in $\mathcal{F}$. So my question is if the automorphism $\phi_p$ can be lifting to $X$. It is if there exist an automorphism $\phi:X\mapsto X$ such that the reduction modulus $p$ of $\phi$ coincide with $\phi_p$. I do this quiestion because if there exist $\phi$ then a solution of $D(x)=0$ is only a eigenvalue of $\phi$. So I have a way to know if this problem has solutions.

  • $\begingroup$ I'm confused, why is $Fr$ invertible? $\endgroup$ – Marc Paul Sep 27 '19 at 21:05
  • $\begingroup$ We can use the etale closure of X. So in positive characteristic I belive that it contain the inseparable closure of the reduction of X in this characteristic. $\endgroup$ – camilo Sep 28 '19 at 0:35
  • $\begingroup$ Have you looked at Lusztig's lifting of a Frobenius map to a quantied enveloping algebra in characteristic 0? $\endgroup$ – Jim Humphreys Sep 28 '19 at 0:52

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