Dualities for locally completely decomposable Abelian groups
All groups here are torsion-free Abelian of finite rank. The literature contains a number of examples of pairs of theorems that clearly have a dual character but were proved independently by ``dualizing proofs''. The examples given in the paper are the following. Theorem 5.1A. Every pure subgroup of a completely decomposable group \(G\) containing no copy of \(\mathbb{Q}\) is again completely decomposable if and only if the typeset of \(G\) is a tree. -- Theorem 5.1B. Every torsion-free epimorphic image of a completely decomposable group \(G\) having no epimorphic image isomorphic with \(\mathbb{Q}\) is again completely decomposable if and only if the typeset of \(G\) is a co-tree. Theorem 5.2A. Let \(G\) be a finite essential extension of a completely decomposable group \(A\). Then \(|G/A|\) is minimal among indices of completely decomposable subgroups if and only if the induced map \(A(\tau)/A^\#(\tau)\to G(\tau)/G^\#(\tau)\) is an isomorphism for each type \(\tau\). -- Theorem 5.2B. Let \(G\) be a subgroup of finite index in a completely decomposable group \(A\). Then \(|A/G|\) is minimal among indices of completely decomposable overgroups of \(G\) if and only if the induced map \(G^\#[\tau]/G[\tau]\to A^\#[\tau]/A[\tau]\) is an isomorphism for each type \(\tau\). Theorem 5.3A. Let \(0\to H\to G\to K\to 0\) be a \({\mathcal K}(n)\)-sequence. If there are fewer than \(n+3\choose 2\) isomorphism classes of rank-1 summands of \(G\), then \(H\) is completely decomposable. -- Theorem 5.3B. Let \(0\to H\to G\to K\to 0\) be a co-\({\mathcal K}(n)\)-sequence. If there are fewer than \(n+3\choose 2\) isomorphism classes of rank-1 summands of \(G\), then \(K\) is completely decomposable. By combining a generalization of Warfield duality (that is too restrictive) with Butler duality (that is too general) the authors establish, with considerable technical effort and expertise, a duality that enables them to derive the above dual results from one another. The many definitions, concepts, and lemmas are too complicated to be given here.
torsion-free Abelian groups of finite rank
scientific article; zbMATH DE number 1021753
scientific article