Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules

TL;DR

This paper develops a generalized singular compactness theorem to characterize $\,\Sigma$-cotorsion modules via their complete theories and provides new descriptions of Gorenstein flat and injective modules, establishing their roles in cotorsion pairs.

Contribution

It introduces a broad singular compactness theorem linking module properties to their theories and offers novel characterizations of Gorenstein modules within cotorsion pairs.

Findings

$\,\Sigma$-cotorsion property is theory-dependent

Gorenstein flat modules form a cotorsion pair

Dual results for Gorenstein injective modules

Abstract

We introduce a general version of singular compactness theorem which makes it possible to show that being a $\Sigma$-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.