The structure of Ext for torsion-free modules of finite rank over Dedekind domains

摘要

The structure of Ext is fundamental to understanding rings and their modules. For example, relative injectivity can be interpreted as a structural property of Ext as in when ExtR(1)(A, B) is torsion-free or zero. The purpose of this work is to examine Ext(R)(1)(A, B) under the assumption that A and B are torsion-free modules of finite rank, and R is a Dedekind domain. It is. the abelian group theorist's mantra that their results extend to modules over Dedekind domains.. This is almost always the case, how ever the structure of Ext is strongly influenced by the rank of the completion of R; unlike the case for the integers, the completion of a domain may have finite rank over the domain. Below, R will represent an integral domain with quotient field Q, and we will assume that R not equal Q. In most case R will be assume to be Dedekind. All unadorned Ext, Hom, and circle times symbols are with respect to R. References: 14