The module of K\"ahler differentials for extensions of valuation rings

TL;DR

This paper characterizes the module of K"ahler differentials for extensions of valuation rings, especially when the ramification index is one, using key polynomials, and determines when this module is trivial.

Contribution

It provides a new characterization of the K"ahler differentials for valuation ring extensions using key polynomials, and identifies conditions for their triviality.

Findings

Characterization of ifferentials in terms of key polynomials

Criteria for when the module of differentials is zero

Application to simple algebraic valued field extensions

Abstract

The main goal of this paper is to characterize the module of K\"ahler differentials for an extension of valuation rings. More precisely, we consider a simple algebraic valued field extension $(L/K,v)$ and the corresponding valuation rings $\VR_L$ and $\VR_K$. In the case when $e(L/K,v)=1$ we present a characterization for $\Omega_{\VR_L/\VR_K}$ in terms of a given sequence of key polynomials for the extension. Moreover, we use our main result to present a characterization for when $\Omega_{\VR_L/\VR_K}=\{0\}$.