On Frobenius and Fibers of Arithmetic Jet Spaces

TL;DR

This paper constructs canonical Frobenius lifts in arithmetic jet spaces and shows that certain kernels form a prolongation sequence, advancing understanding of the structure of arithmetic jet spaces and their fibers.

Contribution

It introduces a method to obtain canonical Frobenius lifts in inverse systems of schemes derived from arithmetic jet spaces, revealing new structural properties.

Findings

Existence of canonical Frobenius lifts in inverse systems of schemes.

The inverse system of kernels of jet space projections forms a prolongation sequence.

Enhanced understanding of the fiber structure 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 $J^*X$ and a prolongation sequence $S^*$ 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 $J^nE \rightarrow E$, then the inverse system $\{N^n\}_n$ is a prolongation sequence.