Fuchs' problem 43
The author proves: Let \(A\), \(C\) be torsion-free abelian groups of finite rank. Then \(\text{Ext}(A,B) \cong \text{Ext}(C,B)\) for every torsion-free group of finite rank \(B\) if and only if \(OT(A) = OT(C)\), \(A = F \oplus A' \oplus D\), \(C = F'\oplus C' \oplus D'\) with \(F\), \(F'\) free groups, \(D\), \(D'\) divisible groups and \(A'\) quasi-isomorphic to \(C'\). Here \(OT(G)\) is the outer type of \(G\), i.e. the supremum of all types of torsion-free rank-one quotients of \(G\). This theorem solves completely a variant of Problem 43 of \textit{L. Fuchs} [Infinite Abelian Groups, Vol. I (Academic Press 1970; Zbl 0209.055)]. The proof uses a theorem of independent interest: For torsion-free groups of finite rank \(A\), \(C\) the following are equivalent. (a) \(A\) is quasi-isomorphic to \(C\). (b) The \(p\)-ranks \(r_ p\text{Hom}(A,B)\) and \(r_ p\text{Hom}(C,B)\) are equal for all primes \(p\) and all torsion-free groups of finite rank \(B\). (c) \(r_ p\text{Hom}(A,B) = r_ p\text{Hom}(C,B)\) for all \(p\) and all torsion-free groups \(B\) whose rank is less or equal to the rank of some quasi-summand of \(A\oplus C\). -- The other component of the proof is contained in another paper by the author [When do two groups always have isomorphic extension groups? Rocky Mt. J. Math. (to appear)] which links the assumption \(\text{Ext}(A,B) \cong \text{Ext}(C,B)\) to parts (b) and (c) of the preceding theorem.
torsion-free abelian groups of finite rank
scientific article; zbMATH DE number 61927