Defects of Algebraic Function Fields, Completion Defects and Defect Quotients

TL;DR

This paper introduces and studies the concepts of completion defect and defect quotient in valued field extensions, providing new insights into the defect phenomenon in valuation theory, especially in positive characteristic.

Contribution

It defines and analyzes the weaker notions of defect, completion defect and defect quotient, extending the understanding of defect in valued function fields and finite extensions.

Findings

Defined completion defect and defect quotient for valued field extensions.

Established analogues of classical defect results for these new notions.

Provided insights into defect behavior in positive characteristic valuation theory.

Abstract

The {\it defect} (also called {\it ramification deficiency}) of valued field extensions is a major stumbling block in deep open problems of valuation theory in positive characteristic. For a detailed analysis, we define and investigate two weaker notions of defect: the {\it completion defect} and the {\it defect quotient}. We define them for finite extensions as well as for certain valued function fields (those with "Abhyankar valuations" that are nontrivial on the ground field). We investigate both defects and present analogues of the results that hold for the usual defect.