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 $J^{\ast }X$ and a prolongation sequence $S^{\ast }$ over the scheme $X$. As a consequence, for any smooth group scheme $E$, if $N^n$ denote the kernel of the canonical projection map of the $n$-th jet space $J^{n}E\to E$, then the inverse system $\{N^n{\}}_n$ is a prolongation sequence.