Kernels of Arithmetic Jet Spaces and Frobenius Morphism

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 $\pi$.