E-Ringe, der Funktor Ext und ein Problem von Laszlo Fuchs
Zusammenfassung der Projektergebnisse
The project focussed on three important topics in the theory of commutative groups: The construction and algebraic properties of non-commutative and commutative E-rings, the classification of decompositions of torsion-free commutative groups into indecomposable summands, and the structure of the group of extensions in various models of set-theory. While the results on E-rings were rather discouraging, several interesting and important results were obtained on the second and third topic. Firstly, E-rings were defined by Phill Schultz in the 1970s as unital rings such that the natural evaluation map between the ring and its underlying commutative group provides an isomorphism. As part of the project E-rings sandwiched between a direct sum of cyclic torsion groups and their corresponding product were studied. However, a suggested technique building towers of rings in order to produce noncommutative E-rings was proven to be inappropriate. Secondly, indecomposable torsion-free groups are the building blocks of torsion-free commutative groups. In the project a new approach to their classification was developed using a particular action of the automorphism group on decompositions into indecomposables. Moreover, existence and uniqueness results on main decompositions of torsion-free commutative groups were shown where a main decomposition of G is a decomposition G = A ⊕ H with A completely decomposable and H clipped, i.e. it has no summands of rank one. On the contrary countable examples were constructed where there is no such decomposition. Finally, a construction method was obtained to realise certain sets of partitions of a natural number n as sequences of ranks of decompositions into indecomposable summands of a fixed torsion-free commutative group. This is related to an old question by Laszlo Fuchs. All these results will have to be examined again for a possible extension to larger ranks in a follow-up project. Thirdly, the structure of groups of extensions has been of interest to algebraists since the 1970s when Saharon Shelah solved the well-known Whitehead problem. He also showed that in certain models of set-theory the group of rational numbers can be realised as Ext(G, Z) for some torsion-free commutative group G. In the project this results was generalised substantially by showing that almost any divisible group of finite rank can be of the form Ext(G, R) for some R ⊆ Q in a suitable forcing extension of the ground model. This result certainly has several applications in e.g. the theory of cellular covers or the theory of tilting and co-tilting modules.
Projektbezogene Publikationen (Auswahl)
- Main Decompositions of Torsion-Free Abelian Groups
G. Braun, P. Schultz and L. Strüngmann
(Siehe online unter https://doi.org/10.1016/j.jalgebra.2019.02.035)
