An extension of Greenberg's theorem to general valuation rings

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.

arXiv comment15 pages, dedicated to Jan Denef; Manuscripta mathematica, 2011. The final article is avaliable at springerlink.com: http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s00229-011-0510-5

Cited by in corpus (5)

• Artin-Mazur-Milne duality for fppf cohomology
• Decidability via the tilting correspondence
• Counter-examples to non-noetherian Elkik's approximation theorem
• Approximation results of Artin-Tougeron-type for general filtrations and for $C^r$-equations
• Immediate extensions of valuation rings and ultrapowers