Generalised Hasse-schmidt Varieties and Their Jet Spaces

Building on the abstract notion of prolongation developed in [10], the theory of iterative Hasse-Schmidt rings and schemes is introduced, simultaneously generalising difference and (Hasse-Schmidt) 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-Schmidt 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-Schmidt variety is determined by its jet spaces at a point.