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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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