Almost flat abelian groups
In a series of lectures at the University of Connecticut at Storrs R. S. Pierce raised in 1989 several questions concerning the impact of flatness of an abelian group \(A\) as a left module over its endomorphism ring \(E(A)\) on \(A\) as group, or \(\mathbb{Q}\otimes_\mathbb{Z} A\) as \(\mathbb{Q}\otimes E(A)\)-module. These questions are answered in the paper under discussion. The investigations lead the authors to say that \(A\) is almost flat as \(E(A)\)-module if \(\text{Tor}^1_{E(A)}(M,A)\) is a bounded abelian group for all right \(E(A)\)-modules \(M\). The class of almost flat \(E(A)\)-modules is shown to be closed under quasi-isomorphism. This theorem seems to be the modified answer intended by Pierce after ruling out counter Example 2.9. It follow characterizations of almost flat \(E(A)\)-modules in case \(A\) has finite rank which are interesting on their own.
scientific article; zbMATH DE number 878056