E-rings, the functor Ext and a problem by Laszlo Fuchs

• Applicant: Professor Dr. Lutz Strüngmann
• Subject Area: Mathematics
• Term: from 2017 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 349703505

The idea of the project is a collaboration between Professor Dr. Phil Schultz from the University of the Western Cape (Australia) and the applicant Strüngmann. The project will focus on three topics from abelian group theory and both applicants are experts in this field.We will investigate properties of groups that are induced by the action of homological functors like Hom(-,-) and Ext(-,-) on them. Firstly, (non-)commutative E-rings, which play an important role in homotopy theory since they are exactly the localisations of the ring of integers, will be studied. Recall that a ring is an E-ring if it is naturally isomorphic to the endomorphism ring of its own additive group. We aim at giving a new construction method for non-commutative E-rings as well as structure theorems for commutative E-rings.A second topic will be properties of the functor Ext in the category of abelian groups. A main focus will be on the question when Ext commutes with direct sums and on the structure of Ext(G,H) in certain models of ZFC. This is related to Shelah's solution of the Whitehead problem.The third topic of the project is the decomposition theory of abelian groups. Here, we will investigate mainly what direct decompositions can be realised for groups. One end is formed by the super decomposable groups, the other end is formed by the indecomposable groups. But what happens in between? Can we prescribe the ranks of the direct summands?All three topics are designed according to the applicants expertise. Both, Schultz and Strüngmann, have been working intensively on E-rings and properties of Ext as well as construction methods for abelian groups. Thus, joining forces should yield to substantial results.

• DFG Programme: Research Grants
• International Connection: Australia
• Cooperation Partner: Professor Dr. Phill Schultz