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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.

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.

Reviewer: C.Vinsonhaler (Storrs)

### MSC:

20K20 | Torsion-free groups, infinite rank |

20K40 | Homological and categorical methods for abelian groups |

20K15 | Torsion-free groups, finite rank |