Logic Workshop

November 9
Philipp Rothmaler, CUNY
Some incomplete theories of modules

Complete theories are in the center of classical model theory, especially in Shelah's classification or stability theory. But there are other, incomplete, theories that rival the complete theories in importance.

For instance, there are the theories whose model classes are closed under direct product, direct limit and pure substructure (a generalization of direct summand). In modules, such model classes are exactly the definable subcategories that arise in representation theory, and they are in bijective correspondence with the closed subsets of the Ziegler spectrum (knowledge of which is not assumed in the talk). But even in a general context, these can be characterized as the classes that are axiomatized by implications of positive primitive formulas (= existentially quantified conjunctions of atomic formulas).

For another instance of implicational (and certainly incomplete) theories, consider the theories of the class of flat modules---or, to bring it down to earth, torsion-free abelian groups---which, over arbitrary rings, require infinitary implications.

Yet another example is that of the theory of all finite abelian groups, whose models are known as pseudofinite abelian groups.

I will discuss the role of such theories in the theory of modules. As an example, I will indicate why every abelian group is a direct summand of a pseudofinite abelian group.