Defects of algebraic function fields, completion defects and defect quotients

Abstract

The defect (also called 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 finer notions of defect: the completion defect and the defect quotient. We define all three defects for finite valued field extensions as well as for certain valued function fields (those with Abhyankar valuations that are allowed to be nontrivial on the ground field). These defects of valued function fields have played an important role in genus reduction formulas that were presented by several authors. We prove the most general known form of the Finiteness and Independence Theorem for the defect of valued function fields. Further, we investigate the completion defect and the defect quotient in detail and present analogues of the results that hold for the usual defect.