Dedekind Domains and the P-rank of Ext

Abstract

We address what can be said of torsion-free finite rank modules $A$ and $B$ over a Dedekind domain $R$ when their Ext's are isomorphic, extending an answer to Fuchs' Problem 43 and its dual by Goeters. We obtain a result for the covariant case when $\hat{R_P}$ has infinite rank over $R$, noting that $A$ and $B$ are quasi-isomorphic iff the $P$-rank of their Hom sets match. In the contravariant case, we see $A$ and $B$ are quasi-isomorphic implies their extension groups are isomorphic, with the converse holding when again $\hat{R_P}$ has infinite rank over $R$. Along the way, we find equivalent conditions that hold for Noetherian domains whose completions are not complete in the $P$-adic topology.