On Positive-Characteristic Semi-Parametric Local-Uniform Reductions of Varieties over Finitely Generated $\mathbb{Q}$-Algebras

TL;DR

This paper proves an equivalence between local characteristic-zero semi-parametric liftings of varieties and their positive-characteristic reductions, facilitating the transfer of algebraic theorems across characteristics.

Contribution

It provides a non-standard proof establishing the equivalence between local characteristic-zero liftings and positive-characteristic reductions with bounded complexity.

Findings

Equivalence between characteristic-zero and positive-characteristic reductions.

Existence of a global complexity bound for reductions.

Facilitates transfer of algebraic theorems across characteristics.

Abstract

We present a non-standard proof of the fact that the existence of a local (i.e. restricted to a point) characteristic-zero, semi-parametric lifting for a variety defined by the zero locus of polynomial equations over the integers is equivalent to the existence of a collection of local semi-parametric (positive-characteristic) reductions of such variety for almost all primes (i.e. outside a finite set), and such that there exists a global complexity bounding all the corresponding structures involved. Results of this kind are a fundamental tool for transferring theorems in commutative algebra from a characteristic-zero setting to a positive-characteristic one.