Generalised Hasse-Schmidt varieties and their jet spaces

TL;DR

This paper develops a unified framework for difference and differential algebraic geometry using iterative Hasse rings and schemes, enabling the study of jet spaces for complex algebraic systems.

Contribution

It introduces a general theory of iterative Hasse rings and schemes, extending jet space constructions to infinite-dimensional algebraic systems.

Findings

Hasse jet spaces constructed for arbitrary algebraic systems

Hasse variety determined by jet spaces under separability

Unified formalism bridges difference and differential algebraic geometry

Abstract

This work provides a unified formalism for studying difference and (Hasse-) differential algebraic geometry, by introducing a theory of "iterative Hasse rings and schemes". 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.