An extension of Greenberg's theorem to general valuation rings

TL;DR

This paper extends Greenberg's strong approximation theorem to a broader class of valuation rings with arbitrary value groups, employing ultraproduct techniques, and applies it to prove a closed image theorem for proper morphisms over valued fields.

Contribution

It introduces a generalized version of Greenberg's theorem for schemes over valuation rings with arbitrary value groups using ultraproducts.

Findings

Extended Greenberg's theorem to general valuation rings.

Proved a closed image theorem for proper morphisms over valued fields.

Demonstrated the applicability of ultraproduct methods in valuation theory.

Abstract

We extend Greenberg's strong approximation theorem to schemes of finite presentation over valuation rings with arbitrary value group, using the ultraproduct method of Becker, Denef, Lipshitz and van den Dries. As an application, we prove a closed image theorem (in the strong topology on rational points) for proper morphisms of varieties over valued fields.