Infinite Abelian groups

On the global dimension of the functor category ((mod R)op, Ab) and a theorem of Kulikov

In the present paper we continue to examine cellular covers of groups, focusing on the cardinality and the structure of the kernel K of the cellular map G ! M. We show that in general a torsion free reduced abelian group M may have a proper class of non-isomorphic cellular covers. In other words, the cardinality of the kernels is unbounded. In the opposite direction we show that if the kernel of a cellular cover of any group M has certain "freeness" properties, then its cardinality is bounded by jM j.

https://www.ems-ph.org/journals/show_pdf.php?issn=1661-7207&vol=1&iss=4&rank=3

On kernels of cellular covers

A Hausdorff topological group $(G,\tau)$ is called locally minimal if there exists a neighborhood $U$ of 0 in $\tau$ such that $U$ fails to be a neighborhood of zero in any Hausdorff group topology on $G$ which is strictly coarser than $\tau.$ Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. 
Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. 
A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudo--convex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. We also prove that a bounded abelian group $G$ admits a non-discrete locally minimal and locally GTG group topology iff $|G|\geq \cont$.

Infinite Abelian groups

Citations

On the global dimension of the functor category ((mod R)op, Ab) and a theorem of Kulikov

On kernels of cellular covers

Locally minimal topological groups

Splitting over nilpotent and hypercentral residuals

Varieties for modules and a problem of steenrod

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