K\"ahler differentials, pure extensions and minimal key polynomials

The main object of study in this paper is the module $\Omega$ of K\"ahler differentials of an extension of valuation rings. We show that in the case of pure extensions $\Omega$ has a very good description. Namely, it is isomorphic to the quotient of two modules defined by final segments in the value group of the valuation. This allows us to describe the annihilator of $\Omega$ and to give criteria for $\Omega$ to be finitely generated and to be finitely presented. We also discuss how to relate the module of K\"ahler differentials of pure extensions to minimal key polynomials.