Assoc. Prof. Dr. Jan Žemlička
Charles University

Self-small products of abelian groups

Let A and B be two abelian groups. The group A is called B-small if the covariant functor Hom(A,-) commutes with all direct sums of the form B(k) and A is a self-small group provided it is A-small. The main aim of the talk is to characterize self-small products applying developed closure properties of the classes of relatively small groups. In particular, we show that a product of a system of abelian groups is self small if and only if it relatively small over a direct sum of the system.

As a consequence of the theory of relatively small groups and the well-known fact that powers Zk of the group Z of all integers is slender for any nonmeasurable cardinal k, we characterize self-small products of finitely generated abelian groups. Namely, the product M of finitely generated groups is self-small if and only if either M is isomorphic to power Zk for some cardinal k, or M is isomorphic to a direct sum of a finitely generated free group F and finite abelian p-groups for each prime number p.

Finally, we also discuss possible application of the developed tools for description of self-compact objects in context of general additive and abelian categories.