Anwendungen von Forcing-Methoden und Logik in der Algebra
Final Report Abstract
This research project focused on the application of methods from logic to algebraic problems. Over the last few decades this has become a powerful tool to show that long standing open problems from algebra are not decidable in daily mathematics based on the axioms of set-theory by Zermelo and Fränkel. In particular, set-theoretic constructions by the method of forcing and prediction principles like Jensen's diamond and box principles or Shelah's black-box were used to prove classification theorems in the category of modules over suitable rings (mostly Abelian groups) and to carry out constructions of modules with desired properties in extension models of the universe. The special fiavor of the project was given by its interdisciplinary character and the interplay between the two fields of algebra and logic. Three topics were specifically addressed: (Derived) cotorsion theories, Butler modules over reasonable rings and symmetry properties of Abelian groups. Cotorsion theories were developed by Salce and later on used to varify the long standing flat cover conjecture by Enochs. Basically, one tries to understand classes X and Y of i?-modules which are maximal with respect to the property that ExtR(G, H) = 0 for all G £ X and H ^ Y, i.e. every extension of a module in Y by some module in X has to split. Obviously, this is related to the well-known Whitehead problem solved by Shelah in the 70s forming the starting point for applying set-theory to algebra. The main outcome of the project are the solutions of two old problems by Baer and Kulikov that could be reduced to the problem of characterizing the pairs (G,T) of torsion-free Abelian groups (modules) G and torsion Abelian groups (modules) T such that Ext(G, T) = 0. In a series of papers (partly in collaboration with Shelah) I was able to give a full characterization for countable groups in ZFC and for uncountable groups in Gödel's constructible universe. Additionally, I carried out constructions of models of set-theory in which such a characterization seems to be impossible. As a byproduct the possible structure of Ext(G, H] was investigated for various groups G and H in suitable extensions of the universe and two open questions on filtration equivalent Abelian groups and dual groups from the standard book on set-theoretic methods in algebra by Eklof and Mekler were answered. The class of Butler modules originated from representation theory of finite partially ordered sets. Butler gave various definitions for countable modules that were later on generalized to the uncountable case in two ways - ßi-modules and ß2-inodules. Today we say that a module G is ßi-Butler if Bext]^{G, T) = 0 for all torsion modules T, i.e. that all balanced exact extensions of G by a torsion module split. A S2-niodule is the union of an ascending chain of pure submodules with successive levels defined by finite rank Butlermodules in the old sense. It had been open for decades whether ßi-modules and ß2-niodules are the same (the latter class is always contained in the former class) until Shelah and I gave a consistent argument for their difference. In the project I extracted the algebraic part of this construction (together with Mader) and studied the endomorphism rings of these creatures. Surprisingly and unexpected in the beginning, it turned out that many rings can be realized as the endomorpliism ring of such a Bi-/not ß2-iTiodule and all attempts to construct counter examples of smallest cardinality as well as attempts to reduce the construction to cardinal arithmetic failed. Finally, the focus of the research was put on the notions of transitivity and full transitivity of modules that go back to Kaplansky. Basically, a module G is (fully) transitive if any two elements x,y E G can be mapped onto each other by an automorphism (endomorphism) whenever there is no obvious reason why not. Together with Goldsmith I studied the notion of weak transitivity that links full transitivity and transitivity for Abelian p-groups and together with Shelah I attacked the main problem by Corner in this area: Is there a fully transitive but non-transitive Abelian p-group with finite first Ulm subgroup? We have succeeded in proving that a counter example must be of size less than or equal to the continuum and there is a conjecture/construction of such a counter example in ZFC. This will hopefully be carried out in summer 2012 and then announced on my homepage at the University of Applied Sciences in Mannheim.
Publications
- Weakly transitive torsion-free abelian groups, Comm. in Algebra 33 (2005), 1177-1191
B. Goldsmith und L. Strüngmann
- A characterization of Ext(G, Z) assuming (V=L), Fundamenta
Mathematicae 193 (2007), 141-151.
S. Shelah und L. Strüngmann
- Ext-universal modules in Gödel's universe, Forum Mathematicum 19 (2007), 307-323.
L. Strüngmann
- On the p-rank of Ext(G, Z) in certain models of ZFC. Algebra and Logic 46 (2007), 200-215.
S. Shelah und L. Strüngmann
- Some transitivity results for torsion abelian groups.
Houston Journal of Mathematics 33 (2007), 941-957.
B. Goldsmith und L. Strüngmann
- A class of finitely Butler groups and their endomorphism rings, Hokkaido Journal of Mathematics 37 (2008), 399-425.
A. Mader und L. Strüngmann
- On endomorphism rings of B1-groups that are not B2-groups when adding
N4 reals to the ground model. Proceedings of AMS 137 (2009), 3657-3668.
L. Strüngmann
- Filtration equivalent N 1-separable abelian groups of cardinality N 1. Annals of Pure and Applied Logic 161 (2010), 935-943.
S. Shelah und L. Strüngmann
- Finite automata representable abelian groups. International
Journal of Algebra and Computation Vol. 21 (8) (2011), 1463-1472.
G. Braun und L. Strüngmann
- The torsion-freeness of Ext for modules over Dedekind
domains. International Journal of Algebra 6 (2012), 399 - 414
U. Albrecfit, S. Friedenberg und L. Strüngmann
