The interplay between algebra and logic
Final Report Abstract
This research project focused on the application of methods from logic, in particular set-theory to algebraic problems and a possible feedback. Over the last few decades set-theory and infinite combinatorics have 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, the method forcing turned out to be highly effective when changing the model of ZFC in a way suitable for the algebraic problem under consideration. The special flavor of the project was given by its interdisciplinary character and the interplay between the two fields of algebra and logic as well as the collaboration between the two research groups from Essen and Münster. Three topics were specifically addressed: Butler modules, the automorphism tower problem, and dual groups. Butler modules: We say that an R-module G is a B1-module if the first Bext-functor over R satisfies Bext(G, T) = 0 for all torsion R-modules T, i.e. all balanced exact extensions of G by a torsion module T split. B2-modules are defined as union of an ascending chain of pure submodules B(alpha) where alpha is an ordinal and where the next step B(alpha+1) is obtained as B(alpha+1) = B(alpha)+ G(alpha) for some finite rank B1-module G(alpha). The classes of Butler modules were originally motivated by results from representation theory of finite partially ordered sets. It is relatively easy to see that the class of B2-modules is contained in the class of B1-modules but it had been open for decades if the converse also holds true. It is now known that the answer is undecidable in ZFC. In this project we studied the endomorphism rings of B1-modules that are not B2-modules in some model as well as an algorithmic approach to finite rank Butler modules. Automorphism tower problem: Given a centreless group G, the group Aut(G) consisting of all automorphisms of G is again centreless, and we may view G as a normal subgroup of Aut(G) by identifying elements of G with the corresponding inner automorphisms. We iterate this process to construct the automorphism tower (G(alpha) | alpha an ordinal) of G by setting G(0) = G, G(alpha+1) = Aut(G(alpha)) and G(tau) = U G(sigma) where sigma ranges over all ordinal less than tau (at successor levels, we identify G(alpha) with the group of inner automorphisms of G(alpha) to obtain an ascending sequence). A result of S. Thomas shows that the automorphism tower of every infinite centreless group of cardinality kappa terminates after less than (2^kappa)+-many steps, i.e. there is an ordinal alpha < (2^kappa)+ with G(sigma) = G(alpha) for all sigma greater than or equal to alpha. The least ordinal with this property is called the height of the automorphism tower of G. In the project new upper bounds for the height of automorphism towers were obtained in various models of ZFC showing the difficulty of the problem to compute these heights explicitly for all groups in a given model. Dual groups: An Abelian group G is called dual group if there exists an Abelian group H such that G =H* = Hom(H,Z). Dual groups have a long history and were studied extensively in the literature. For instance, a complete chapter in the standard book by Eklof and Mekler on set-theoretic methods in algebra is dedicated to dual groups and their properties. Easiest examples are the Baer Specker group Z^omega which is the dual of Z^(omega) and conversely, Z^(omega) as the dual of the Baer Specker group. Moreover, many dual groups can be obtained as reflexive groups, i.e. groups for which the natural map from G into its double dual Hom(G*,Z) is an isomorphism. In the project we solved three questions from the book by Eklof and Mekler that had been open for some time: (I) There is a pure non-reflexive non dual subgroup of the Baer-Specker group of cardinality the continuum (II) Consistently Z^(omega) has a dual subgroup of the first uncountable cardinality and the continuum is arbitrarily large (III) Consistently there is a subgroup H of Z(^omega) such that the n-th dual of H is not isomorphic to the (n+1)-th dual of any other group G.
Publications
- Automorphism towers and definability in generalized Baire spaces. Dissertation. Westfalische Wilhelms-Universitat Munster, 2012.
P. Lucke
- Iteratively changing the heights of automorphism towers. Notre
Dame Journal for Formal Logic, Vol. 53, Issue 2 (2012), pp. 155-174.
G. Fuchs and P. Lucke
(See online at https://doi.org/10.1215/00294527-1715662) - The torsion-freeness of Ext for modules
over Dedekind domains, International Journal of Algebra Vol. 6, 2012, pp. 399-414.
U. Albrecht, S. Friedenberg, L. Strungmann
(See online at https://doi.org/10.1215/00294527-1715662) - Σ11-definability at uncountable regular cardinals.
Journal of Symbolic Logic, Vol. 77, Issue 3, 2012, pp. 1011-1046.
P. Lucke
(See online at https://dx.doi.org/10.2178/jsl/1344862172) - (Cellular covers of cotorsion-free modules) ℵ n -FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL. Glasgow Mathematical Journal, 55. 2013, Issue 2, pp 369-380.
R. Gobel, J. Rodriguez and L. Strungmann
(See online at https://doi.org/10.1017/S0017089512000614)
