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

arXiv comment34 pages; small changes made and details added

References in corpus (3)

• On exact categories and applications to triangulated adjoints and model structures
• Periodic modules and acyclic complexes
• The countable Telescope Conjecture for module categories

Cited by in corpus (19)

• Fp-projective periodicity
• $n$-Cotorsion pairs
• Projectively coresolved Gorenstein flat dimension of groups
• Flat relative Mittag-Leffler modules and approximations
• Finiteness criteria for Gorenstein flat dimension and stability
• Model structures, n-Gorenstein flat modules and PGF dimensions
• Locally coherent exact categories
• Cut cotorsion pairs
• Gorenstein modules and dimension over large families of infinite groups
• Auslander conditions and tilting-like cotorsion pairs
• Flat model structures and Gorenstein objects in functor categories
• Finiteness criteria for Gorenstein homological dimension and some invariants of groups
• Generalized periodicity theorems
• Relative Gorenstein flat modules and Foxby classes and their model structures
• Gorenstein analogues of a projectivity criterion over group algebras
• On some triangulated categories over group algebras
• Intersection of complete cotorsion pairs
• Cotilting with balanced big Cohen-Macaulay modules
• The rigidity of filtered colimits of n-cluster tilting subcategories