Kernel of arithmetic jet spaces

Fix a Dedekind domain O and a non-zero prime p in it along with a uniformizer ?. In the first part of the paper, we construct m-shifted ?-typical Witt vectors Wmn(B) for any O algebra B of length m+n+1. They are a generalization of the usual ?-typical Witt vectors. Along with it we construct a lift of Frobenius, called the lateral Frobenius F~:Wmn(B)?Wm(n?1)(B) and show that it satisfies a natural identity with the usual Frobenius map. Now given a group scheme G defined over Spec R, where R is an O-algebra with a fixed ?-derivation ? on it, one naturally considers the n-th arithmetic jet space JnG whose points are the Witt ring valued points of G. This leads to a natural projection map of group schemes u:Jm+nG?JmG. Let NmnG denote the kernel of u. One of our main results then prove that for n?1, NmnG is naturally isomorphic to Jn?1(Nm1G) as group schemes. Hence this implies that for any ?-formal group scheme G^ over Spf R, NmnG^ is isomorphic to Jn?1(Nm1G). As an application, if G^ is a smooth commutative ?-formal group scheme of dimension d and R is of characteristic 0 whose ramification is bounded above by p?2, then our result implies that JnG is a canonical extension of G^ by (Wn?1)d where Wn?1 is the ?-formal group scheme A^n endowed with the group law of addition of Witt vectors. Our results also give a geometric characterization of G(?n+1R) which is the subgroup of points of G(R) that reduces to identity under the modulo ?n+1 map.