# Lift the relative Frobenius automorphism to zero characteristic

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.

• I'm confused, why is $Fr$ invertible? – Marc Paul Sep 27 '19 at 21:05
• 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. – camilo Sep 28 '19 at 0:35
• Have you looked at Lusztig's lifting of a Frobenius map to a quantied enveloping algebra in characteristic 0? – Jim Humphreys Sep 28 '19 at 0:52