Generalised Hasse Varieties and Their Jet Spaces

Building on the abstract notion of prolongation developed in [7], the theory of iterative Hasse rings and schemes is introduced, simultaneously generalising difference and (Hasse-)differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related geometries. As an application, Hasse jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse variety is determined by its jet spaces at a point.