Duality and self-reflexive torsion-free Abelian groups
Given an abelian group \(A\), let \(M_A\) denote the category of those abelian groups that are isomorphic to \(\text{End}(A)\)-submodules of finite direct sums of copies of \(A\). For \(A\) a rank-one torsion-free abelian group, R. B. Warfield jun. showed that the functor \(\text{Hom}(-,A)\) defines a rank-preserving duality on \(M_A\) (all unadorned Hom's are over the integers, \(\mathbb{Z}\)). In this paper the results of Warfield are extended by determining the torsion-free abelian groups \(G\) of finite rank for which \(\text{Hom}(-,G)\) defines a rank-preserving duality on \(M_G\). In particular, the author investigates those \(G\) for which \(G\cong\text{Hom}(\text{Hom}(G,G),G)\), that is, for which \(G\) is self-reflexive. Assume \(G\) is self-reflexive. Then \(\text{End}(G)\) is an \(E\)-ring: \(R=\text{End}(G)\) satisfies \(R\cong\text{End}(R,+)\). Moreover, \(G\) is an \(E\)-module over \(R\); \(\text{Hom}(R,G)=\text{Hom}_R(R,G)\). The structure of \(E\)-rings is well-known and can be used to obtain a decomposition theorem for self-reflexive groups. Corollary 1. Let \(G\) be torsion-free with endomorphism ring \(R\) that is of finite rank. Then \(G\) is self-reflexive if and only if \(G\) quasi-decomposes as \(G_1\oplus G_2\oplus\cdots\oplus G_n\), with each \(G_j\) strongly indecomposable and an \(E\)-module over \(R\), and \(\text{Hom(End}(G_i),G_j)=0\) for \(i\neq j\). A number of methods for constructing examples of self-reflexive groups are presented in Section 2 of the paper. Section 3 is an investigation of the \(G\)-reflexive groups, where \(G\) is a fixed self-reflexive group with finite rank endomorphism ring \(R\). A group \(M\) is \(G\)-reflexive if \(M\) is naturally isomorphic to \(\text{Hom(Hom}(M,G),G)\). Since \(R \) is an \(E\)-ring, \(R\) is quasi-equal to a product of subrings of algebraic number fields and we can form the integral closure \(S\) of \(R\) in \(QR\). It is well-known that \(S/R\) is finite. Let \(P_G\) be the category of torsion-free abelian groups \(M\) of finite rank such that \(M\) is an \(R\)-submodule of a product of copies of \(G\) that is \(p\)-pure whenever \(p\) is a prime dividing \(|S/R|\). Corollary 3. Let \(G\) be self-reflexive of rank-one over \(R=\text{End}(G)\). The following are equivalent for a finite rank torsion-free abelian group \(M\). (a) \(M\) is \(G\)-reflexive. (b) \(M=\text{Hom}(K,G)\) for some \(K\). (c) \(M\) is a \(\mathbb{Z}\)-pure \(R\)-submodule of a product of copies of \(G\). (d) \(M\) is an object in \(P_G\). The main theorem of the paper is the following. Theorem 6. Let \(G\) be torsion-free of finite rank. The following are equivalent. (a) \(\text{Hom}(-,G)\) defines a rank-preserving duality on \(M_G\). (b) \(G\) is self-reflexive of \(\text{End}(G)\)-rank one and \(P_G=M_G\). (c) \(R=\text{End}(G)\) is an \(E\)-ring with \(R\)-rank of \(G\) equal one, and the functor \(\text{Hom}_R(-,G)\) is exact on the category of \(\text{End}_R(G)\)-submodules of finite direct sums of copies of \(G\). (d) \(\text{Hom}(-,G)\) defines an exact rank-preserving duality on \(M_G\).
torsion-free Abelian groups of finite rank
scientific article; zbMATH DE number 1308352
scientific article