Kernels of Arithmetic Jet Spaces and Frobenius Morphism

TL;DR

This paper explores the structure of arithmetic jet spaces of $ ext{pi}$-formal group schemes, showing how Frobenius morphisms induce natural ring maps between Witt vectors, generalizing classical Witt vector operations.

Contribution

It introduces a new perspective on the Frobenius morphism in arithmetic jet spaces, establishing its relation to Witt vector maps and generalizing known structures for formal groups.

Findings

Frobenius morphism restricts to kernels of projection maps.

Induces natural ring maps between shifted Witt vectors.

Generalizes classical Witt vector operations to formal group schemes.

Abstract

For any $\pi$-formal group scheme $G$, the Frobenius morphism between arithmetic jet spaces restricts to generalized kernels of the projection map. Using the functorial properties of such kernels of arithmetic jet spaces, we show that this morphism is indeed induced by a natural ring map between shifted $\pi$-typical Witt vectors. In the special case when $G = \hat{\mathbb{G}}_a$, the arithmetic jet space, as well as the generalized kernels are affine $\pi$-formal planes with Witt vector addition as the group law. In that case the above morphism is the multiplication by $\pi$ map on Witt vector schemes. In fact, the system of arithmetic jet spaces and generalized kernels of any $\pi$-formal group scheme $G$ along with their maps and identitites satisfied among them are a generalization of the case of the Witt vector scheme with the system of maps such as the Frobenius, Verschiebung and multiplication by $\pi$.