Model theory of modules

Cuny Bronx Community College, Bronx NY

Investigators

Abstract

The investigators intend to apply model-theoretic methods to three areas of module theory. The first is the decomposition theory of projective modules and pure-projective modules, where realizing and omitting finitely generated positive primitive types one can obtain and classify interesting new examples of projective and pure-projective modules. In the second area, of cotorsion theories, the investigators intend to establish and apply the interplay with the Ziegler spectrum in order to describe the flat cover and the realm of cotorsion modules that are not pure-injective. The third part is about calculation of Ziegler spectra of string algebras as a contribution to representation-theoretic classification and to decidability theory and about the related issue of super-decomposable pure-injectives. As a fourth, more general model-theoretic topic of study, the investigators propose to apply classical model-theoretic constructions to pairs of structures endowed with an epimorphism in order to obtain existence results in `dual' or `projective' model theory (dual in the sense that monomorphisms are replaced by epimorphisms). The model-theoretic study of mathematical structures is unique in that one restricts to a certain formal language, similar to a computer language. The loss of richness of expression allows for a gain on the side of the structures under investigation: more structures come into play, because the language cannot distinguish between them. (To persue the analogy with computers: a computer may not be able to distinguish between an infinite object and a large finite chunk of it.) The advantage of this approach is that one discovers new structures (models of the formal statements describing the initial structure) that may be very similar to the original structure at hand when the statements mentioned are chosen as to force them to be. It has found striking applications in various parts of mathematics over the last decades.