Defects of Algebraic Function Fields, Completion Defects and Defect Quotients

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.