Duality in some classes of torsion-free Abelian groups of finite rank
Let \(\sigma\), \(\tau\) be a pair of types of torsion-free (abelian) groups of rank 1 which are determined by characteristics \((k_ p)\), \((m_ p)\) such that \(k_ p\leq m_ p\) for all primes \(p\). A torsion-free group \(A\) of finite rank \(n\) belongs to the class \(D^{\tau}_{\sigma}\) iff there exists a free subgroup \(J\) of rank \(n\) of \(A\) such that any \(p\)-primary component of \(A/J\) is a direct sum of \(n\) groups isomorphic either to \(Z(p^{k_ p})\) or to \(Z(p^{m_ p})\). For primes \(p\) such that \(k_ p<m_ p\) let \(s_ p(A)\), \(t_ p(A)\) denote the number of isomorphic summands so \((s_ p(A),t_ p(A))\) are determined up to an equivalence (similar to that of characteristics). It is easy to see that the type of any element of any group in \(D^{\tau}_{\sigma}\) is greater than or equal to \(\sigma\). A detailed study of groups \(A/p^{m_ p}A\) and of \(p\)-adic completions of \(A\) leads to a notion of duality in the class \(D^{\tau}_{\sigma}\), i.e. a contravariant functor \(-^*: D^{\tau}_{\sigma}\to D^{\tau}_{\sigma}\) (where \(D^{\tau}_{\sigma}\) is considered as the category of groups A with a fixed basis of \(J\) and with quasihomomorphisms as maps) such that 1) rank \(A^*=rank A\), 2) \(s_ p(A^*)=t_ p(A)\), \(t_ p(A^*)=s_ p(A)\) for almost all \(p\), 3) \(A^{**}=A\), \(\phi^{**}=\phi\) for all groups \(A\) and all quasihomomorphisms \(\phi\). Any two such dualities are equivalent. This duality is compared with those defined by Warfield and Arnold; e.g. it is shown that if \(m_ p=\infty\) implies \(k_ p=\infty\) then \(-^*=\Hom(-,R)\) where \(R\) is a torsion free group of rank 1 and of type \(\sigma+\tau\).
scientific article; zbMATH DE number 4029832