Quasi-co-local subgroups of Abelian groups. (Q995598)

The two authors investigate a new concept that is closely related to the concepts of cellular covers and co-local-subgroups of Abelian groups -- concepts that have attracted a lot of attention recently in the theory of Abelian groups. In this paper the authors deal with Abelian groups but consider the quasi-category, this is to say, the objects under consideration are Abelian groups \(A,B\) but the homomorphisms are elements of the set \(\mathbb{Q}\otimes\Hom(A,B)\). A subgroup \(K\) of some Abelian group \(G\) is called quasi-co-local if the short exact sequence \(0\to K\to G@>\pi>>G/K\to 0\) induces an isomorphism \(\mathbb{Q}\otimes\Hom(G,G) @>\text{id}_\mathbb{Q}\otimes\pi^*>>\mathbb{Q}\otimes\Hom(G,G/K)\). Obviously, any co-local subgroup of an Abelian group \(G\) is also quasi-co-local. The authors prove various nice results on quasi-co-local groups showing similarities and differences between co-local subgroups and quasi-co-local subgroups. For instance, if \(G\) is a torsion group, then a subgroup \(K\) of \(G\) is quasi-co-local if and only if \(K\) is bounded (Theorem 2.12). In the torsion-free case, the situation is more interesting. For example, if \(A\) is a pure subgroup of the \(p\)-adic integers \(J_p\) and \(K\) is a non-zero subgroup of \(A\), then \(K\) is quasi-co-local in \(A\) if and only if \(A\) is isomorphic to a finite rank pure subring of \(J_p\) and \(K\) is a pure, co-rank \(1\) subgroup of \(A\) (Theorem 3.4). Moreover, if \(G\) is torsion-free of finite rank and \(\text{rk}(G)=\text{rk}(\text{End}(G))\) and \(\mathbb{Q}\text{End}(G)\) is a division ring, then any pure co-rank \(1\) subgroup \(K\) of \(G\) is quasi-co-local in \(G\) (Theorem 3.6). Finally, the authors use a Black-Box construction to prove that for any cotorsion-free group \(L\) and subgroup \(K\) of \(L\) such that \(L/K\) is reduced torsion, there exist arbitrarily large cotorsion-free groups \(G\) such that \(L\) is a co-local subgroup of \(G\) and \(K\) is a quasi-co-local subgroup of \(G\). The construction leads to an open question, namely if there is a cotorsion-free group \(G\) with a quasi-co-local subgroup \(K\) such that its purification \(K_*\) is not a quasi-co-local subgroup of \(G\)?