Applications of model theory to cotorsion modules

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

The theory of purity that arises in abelian group theory and, more generally, module theory may be developed (Crawley-Boevey) in the setting of a locally finitely presented category. The prototypical example of such a category is the category of flat (left) modules over a ring R. The pure-injective objects of the category of flat R-modules are the flat cotorsion modules. The existence of cotorsion envelopes in the category of left R-modules is a consequence of the Flat Cover Conjecture (proved by Bican, El Bashir and Enochs). The project proposes to develop a theory of flat cotorsion modules that generalizes the classical theory of pure-injective modules over R. Ring Theory is the area of Algebra that concentrates on the interaction between possible notions of multiplication and addition. The representation theory of rings, better known as Representation Theory, is devoted to the study of modules. These mathematical structures generalize the concept of a vector space and provide a medium where such instances of multiplication and addition may manifest themselves. The Model Theory of Modules proposes to understand modules from the point of view of Mathematical Logic. Thus one is not only interested in solving linear equations, but also in deciding when a sentence in the appropriate formal language becomes true when it is assigned a given interpretation. The modules that arise most naturally from this point of view (and are the subject of this project) are the pure-injective, or algebraically compact, modules. They also occur prominently in the classical (non-logical) approach to Representation Theory and so provide fruitful ground for further interaction between Algebra and Mathematical Logic.