Twisted Galois stratification

TL;DR

This paper establishes a direct image theorem for Galois formulas in difference schemes, enabling effective quantifier elimination and a detailed algebraic-geometric description of definable sets over fields with Frobenius automorphisms.

Contribution

It introduces a new direct image theorem for Galois formulas and provides an effective quantifier elimination method in the context of difference schemes.

Findings

Proved a direct image theorem for Galois formulas.

Developed an effective quantifier elimination procedure.

Provided a geometric description of definable sets over fields with Frobenius.

Abstract

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulae associated with finite Galois covers of difference schemes.