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Duality in some classes of torsion-free Abelian groups of finite rank. (English. Russian original) Zbl 0633.20031
Sib. Math. J. 27, 563-571 (1986); translation from Sib. Mat. Zh. 27, No. 4(158), 117-127 (1986).
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 $$-^*=\operatorname{Hom}(-,R)$$ where $$R$$ is a torsion free group of rank 1 and of type $$\sigma+\tau$$.
Reviewer: St.Balcerzyk

##### MSC:
 20K15 Torsion-free groups, finite rank 20K40 Homological and categorical methods for abelian groups
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