## An extension of Warfield duality for abelian groups.(English)Zbl 0845.20042

In 1968 R. B. Warfield, jun. showed how to adapt vector space duality to torsion-free abelian groups (hereafter, simply “groups”) of finite rank. If $$A$$ is a subgroup of the additive rationals, a group $$M$$ is called $$A$$-locally free if $$M$$ embeds as a subgroup of a (finite) product of copies of $$A$$ and for each prime $$p$$, $$pA=A$$ implies $$pM=M$$ ($$M$$ is an $$\text{End}(A)$$-module). Warfield duality says that $$\text{Hom}(-,A)$$ is a duality on the $$A$$-locally free groups. In this note, Warfield duality is extended by allowing $$A$$ to be more general. The author defines a class $$\Gamma(p)$$ to be the groups $$G$$ with the following properties: (1) $$p\text{-rank}(G)=1$$; and $$G$$ is $$p$$-reduced; (2) $$G$$ is a rank-1 module over $$R=\text{End}(G)$$; (3) A group $$M$$ is naturally isomorphic to $$\text{Hom}_R(\text{Hom}_R(M,G),G)$$ precisely when $$M$$ is an $$\text{End}_R(G)$$-submodule of a product of copies of $$G$$.
Note that no assumption is made on rank. Any group $$G$$ satisfying (1) and (2) is homogeneous and $$\text{End}_Z(G)$$ is an integral domain. On the other hand, by a result of Arnold, O’Brien and Reid, any homogeneous indecomposable finite rank group which is quasi-pure injective is in $$\Gamma(p)$$ for some $$p$$. Since the author intends to use $$\text{Hom}(-,G)$$, for $$G\in\Gamma(p)$$ as a duality functor, condition (3) seems artificial. However, that condition is not needed for groups of finite rank.
Theorem. Let $$G$$ be a torsion-free group of finite rank satisfying (1) and (2). Then a finite rank abelian group $$M$$ is naturally isomorphic to $$\text{Hom}_Z(\text{Hom}_Z(M,G),G)$$ if and only if $$M$$ is an $$\text{End}(G)$$ module and embeds as a pure subgroup of a product of copies of $$G$$.
The “locally-free” result in the general case is the following Corollary. Let $$G\in\Gamma(p)$$ and let $$M$$ be an abelian group of finite $$p$$-rank. Then $$M\simeq\text{Hom}_Z(K,G)$$ for some abelian group $$K$$ if and only if $$M$$ embeds as a subgroup of a product of copies of $$G$$ and $$M$$ is a left $$\text{End}(G)$$ module. The author illumines these and other duality theorems with numerous examples. A final section in the paper investigates an injective lifting property that is related to the duality results.

### MSC:

 20K20 Torsion-free groups, infinite rank 20K40 Homological and categorical methods for abelian groups 20K15 Torsion-free groups, finite rank
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