Arithmetic jet spaces

One of the primary objectives of this paper is to establish compatibility between two different jet space functors in the most general context. We first show an adjunction between the jet and the Witt functor on algebras. Following Borger's approach, we then construct the algebraic jet space functor in the general setting where the base is an arbitrary prolongation sequence. We then show that this functor is representable in the category of schemes and that Buium's jet space can be recovered by the π-adic completion of this representable scheme. As an application, this allows us to strengthen a result of Buium on the relation between Greenberg's transform and the special fiber of jet spaces, including the ramified case.