Generalized Hasse-Schmidt varieties and their jet spaces

Abstract

Building on the abstract notion of prolongation developed by Moosa and Scanlon (‘Jet and prolongation spaces’, J. Inst. Math. Jussieu 9 (2010), 391–430), the theory of iterative Hasse–Schmidt rings and schemes is introduced, simultaneously generalizing 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.

Existentially closed fields withG-derivations

“…It is not clear for us what happens for the other formal groups. We also discuss the notion of "canonical G-tuples" (see Definition 6.1), which generalizes Ziegler's notion of canonical p-basis [32], and possible generalizations of our methods to the context of [21].…”

Model theory of fields with virtually free group actions

“…The common context between ‐fields and fields with iterative Hasse–Schmidt derivations (that is, also differential fields) would be fields with algebraic actions of a fixed formal group scheme. A general theory of such actions is considered in for example, . However, we would rather not state any conjecture for the existence of a model companion in such a general case.…”

A note on extending actions of infinitesimal group schemes

“…(iii) For a more general context (including and going beyond HS-derivations), the reader is advised to consult Example 2.12 in [18].…”