On Frobenius and Fibers of Arithmetic Jet Spaces

Abstract: In this article, given a scheme $X$ we show the existence of canonical lifts of Frobenius maps in an inverse system of schemes obtained from the fiber product of the canonical prolongation sequence of arithmetic jet spaces $J<sup>*X$ and a prolongation sequence $S<sup>*$ over the scheme $X$. As a consequence, for any smooth group scheme $E$, if $N<sup>n$ denote the kernel of the canonical projection map of the $n$-th jet space $J^nE \rightarrow E$, then the inverse system ${N<sup>n}_n$ is a prolongation sequence.