Kähler differentials, pure extensions and minimal key polynomials

Abstract

International audienceThe main object of study in this paper is the module Ω of Kähler differentials of an extension of valuation rings. We show that in the case of pure extensions Ω has a very good description. Namely, it is isomorphic to the quotient of two fractional ideals of the value group of the valuation. This allows us to describe the annihilator of Ω and to give criteria for Ω to be finitely generated and to be finitely presented. We also discuss how to relate the module of Kähler differentials of pure extensions to minimal key polynomials