Department of Mathematics
Server

I am working my way through a fascinating new paper [1] which relates some of the "hottest" new ideas in model theory and in module theory in a substantial way. In these two talks I will give an informal account of the definitions of these concepts, and an outline of the main results.

An "abstract elementary class" is, loosely speaking, a category that generalizes the idea of the class of models of a theory together with the elementary embeddings between them. For a module N, perp-N is the set of all modules A such that Extⁱ(A,N)=0 for all i≥1.

The main results of the paper investigate when perp-N yields an abstract elementary class; this turns out to be related to standard ideas of the model theory of modules. The authors state that "The surprising fact is that some of the basic model theoretic properties of the class [induced by] perp-N translate directly into algebraic properties of the class perp-N and the module N over the ring R that have previously been studied in quite a differen context (approximation theory of modules, infinite dimensional tilting theory, etc)."

[1] perp-N as an Abstract Elementary Class, John T. Baldwin, Paul C. Eklof, and Jan Trlifaj. Preprint, Dec. 28, 2006.