On defect in finite extensions of valued fields
Caio Henrique Silva de Souza, Mark Spivakovsky
TL;DR
The defect of finite extensions of valued fields is a central obstacle in problems like local uniformization. The paper analyzes how three properties—simply defectless, immediate-defectless, and algebraically maximal—relate to each other, establishing that simply defectless implies algebraically maximal and that immediate-defectless is equivalent to the henselization being algebraically maximal; it also provides a defectless-but-not-algebraically-maximal example. The main contribution is a construction of an algebraically maximal field that admits a simple defect extension, achieved via the framework of $w$-quasi-finite elements in generalized power series fields. These results clarify the boundaries between defectless and non-defectless behavior in valued fields and have implications for local uniformization and related model-theoretic questions.
Abstract
In recent decades, the defect of finite extensions of valued fields has emerged as the main obstacle in several fundamental problems in algebraic geometry such as the local uniformization problem. Hence, it is important to identify defectless fields and study properties related to defect. In this paper we study the relations between the following properties of valued fields: simply defectless, immediate-defectless and algebraically maximal. The main result of the paper is an example of an algebraically maximal field that admits a simple defect extension. For this, we introduce the notion of quasi-finite elements in the generalized power series field $k\left(\left(t^Γ\right)\right)$.