We introduce a general version of the 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.

中文翻译：

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$$ \ Sigma $$Σ-cotorsion和Gorenstein模块的奇异紧致性和可定义性

我们介绍了奇异紧致性定理的一般形式，这使得有可能证明作为\（\ Sigma \）-扭曲模块是该模块完整理论的一个属性。作为沿途开发的功能强大的工具的应用，我们对Gorenstein扁平模块进行了新的描述，这意味着，不管环是什么，所有Gorenstein扁平模块的类别都构成了一个完美的扭矩对的左手类别。我们还证明了Gorenstein内射模的双重结果。