Prolongations in differential algebra

We develop the theory of higher prolongations of algebraic varieties over fields in arbitrary characteristic with commuting Hasse-Schmidt derivations. Prolongations were introduced by A. Buium in the context of fields of characteristic 0 with a single derivation. We give a new construction of higher prolongations in a more general context, inspired by work of P. Vojta. Generalizing a result of Buium in characteristic 0, we prove that these prolongations are represented by a certain functor, which shows that they can be viewed as ‘twisted jet spaces’. We give a new, constructive proof of a theorem of R. Moosa, A. Pillay, and T. Scanlon that the prolongation functor and jet space functor commute. We also give a direct, combinatorial proof of the fact that the m prolongation and m jet space are differentially isomorphic by showing that their infinite prolongations are isomorphic as schemes. We develop the theory of prolongations of schemes over fields with finitely many commuting HasseSchmidt (or ‘higher’) derivations in arbitrary characteristic. Prolongations over a differential field of characteristic 0 were introduced by Buium [Bui92], but have also been considered in more general contexts [BV95, BV96, Sca97, MPS04]. In an appendix, Buium observes that his prolongations can be viewed as ‘twisted’ jets spaces, by considering the functors they represent. When the derivation is trivial, then his prolongations are just the usual jet spaces. In this more general context, we give a direct construction of prolongations, and also show that they represent the natural generalization of the functor of Buium to the context of commuting higher derivations. There are a few reasons to consider higher derivations rather than the usual derivations. Our constructions are adapted from Vojta’s construction of jets via higher derivations [Voj04], which makes it easier to work in this context. In characteristic 0, there is no real difference, as derivations and higher derivations are interdefinable. This is no longer true in positive characteristic, but here it seems that higher derivations may be more important, at least for model-theoretic applications [Hru96, Sca97]. Good general references for the model theory of differential fields are [Mar96, Pil02, Sca02]. For applications to diophantine geometry, see, for example, [HP00, PZ03, Pil04]. For expository reasons, we develop the theory first for a single higher derivation, and then for commuting derivations. Many of the definitions, results, and proofs for a single derivation will adapt straightforwardly to the general case. 1 Higher derivations Definition 1.1. (See [Mat89] or [Voj04].) Let R be a ring, f : R → A and R → B be R-algebras, and m ∈ N ∪ {∞}. A higher derivation of order m from A to B over R is a sequence D = (D0, . . . , Dm), or (D0, . . .) if m = ∞, where D0 : A → B is an R-algebra homomorphism and D1, . . . , Dm : A → B are homomorphisms of (additive) abelian groups such that 1. Di(f(r)) = 0 for all r ∈ R and i ≥ 1;