Kernels of Arithmetic Jet Spaces and Frobenius Morphism

For any $π$-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 $π$-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 $π$-formal planes with Witt vector addition as the group law. In that case the above morphism is the multiplication by $π$ map on Witt vector schemes. In fact, the system of arithmetic jet spaces and generalized kernels of any $π$-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 $π$.

💡 Research Summary

The paper investigates the interaction between Frobenius morphisms and arithmetic jet spaces attached to a π‑formal group scheme G over a π‑adic base. Starting from Buium’s δ‑geometry, the authors fix a Dedekind domain O with a non‑zero prime ideal π, a π‑adic complete O‑algebra R equipped with a π‑derivation δ, and consider π‑formal schemes over S = Spf R. For any integer n≥0 the n‑th arithmetic jet functor JⁿX is defined as the functor of points X(Wₙ(–)), where Wₙ denotes the π‑typical Witt vectors of length n+1. The usual projection maps uₘ: J^{m}{n}G → J^{m‑1}{n}G are available for all m,n≥0.

The authors introduce the “generalized kernel” N_{r,m}^{s,n}(G) as the kernel of uₘ, but they describe it functorially using shifted Witt vectors. For a fixed shift m they define the ring of m‑shifted Witt vectors of length m+n+1, \