Kernel of arithmetic jet spaces

Abstract

Fix a Dedekind domain O and a non-zero prime p in it along with a uniformizer π. In the first part of the paper, we construct m-shifted π-typical Witt vectors Wmn(B) for any O algebra B of length m+n+1. They are a generalization of the usual π-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 π-derivation δ 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 π-formal group scheme G^ over Spf R, NmnG^ is isomorphic to Jn−1(Nm1G). As an application, if G^ is a smooth commutative π-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 π-formal group scheme A^n endowed with the group law of addition of Witt vectors. Our results also give a geometric characterization of G(πn+1R) which is the subgroup of points of G(R) that reduces to identity under the modulo πn+1 map.