The defect formula

TL;DR

This paper characterizes the defect of simple algebraic extensions of valued fields, extending known results for henselian fields, using graded algebra tools to analyze valuations and their extensions.

Contribution

It generalizes the defect characterization to non-henselian cases and establishes an isomorphism between graded algebras for valuations on henselizations.

Findings

The defect equals the product of relative degrees of limit augmentations in the generalized setting.

The graded algebra map induced by valuation extension is an isomorphism.

Provides a new framework for understanding defects in algebraic extensions.

Abstract

In this paper we present a characterization for the defect of a simple algebraic extensions of valued fields. This characterization generalizes the known result for the henselian case, namely that the defect is the product of the relative degrees of limit augmentations. The main tool used here is the graded algebra associated to a valuation on a polynomial ring. Let $\kh$ be a henselization of a valued field $K$. Another relevant result proved in this paper is that for every valuation $\muh$ on $\khx$, with restriction $\mu$ on $\kx$, the corresponding map $\mathcal G_\mu\hk\mathcal G_{\muh}$ of graded algebras is an isomorphism.