2022/04/24 by Arnab Saha, Saha, Arnab
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2204.11250
openalex publication_date 2022/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Since the results here have been superseded by another paper cowritten by the author, this article is available for reference purposes only. Fix a Dedekind domain O and a non-zero prime \mathfrakp 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 imply 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 (\mathbbWn-1)d where \mathbbWn-1 is the π-formal group scheme \mathbbAn 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.