Singular compactness and definability for Sigma-cotorsion and Gorenstein modules

Abstract

We introduce a general version of the singular compactness theorem which makes it possible to show that being a -cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair.

We also prove the dual result for Gorenstein injective modules.

Keywords

• Singular compactness
• definability
• Sigma-cotorsion
• Gorenstein modules