Discriminant as a product of local discriminants

Abstract

Let 𝑅 be a discrete valuation ring with maximal ideal 𝔭 and 𝑆 be the integral closure of 𝑅 in a finite separable extension 𝐿 of 𝐾. For a maximal ideal 𝔓 of 𝑆, let 𝑅ˆ𝔭,Ŝ𝔓 denote respectively the valuation rings of the completions of 𝐾,𝐿 with respect to 𝔭,𝔓. The discriminant satisfies a basic equality which says that disc(𝑆/𝑅)𝑅ˆ𝔭=∏𝔓∣∣𝔭disc(Ŝ𝔓/𝑅ˆ𝔭). In this paper, we extend the above equality on replacing 𝑅 by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.