Applications of model theory to cotorsion modules

在阿贝尔群理论中出现的纯度理论，以及更普遍的模理论，可以在局部有限表示范畴的背景下发展（克劳利-博维）。这类范畴的原型例子是环R上的左模范畴。平坦R-模范畴的纯内射对象是平坦余挠模。左R-模范畴中余挠子的存在性是平覆盖猜想（由Bican，El Bashir和Enochs证明）的一个推论。该项目提出发展一个理论oflat余挠模，推广了经典理论的纯内射modulesover R.环理论是代数领域，集中在乘法和加法的可能性之间的相互作用。环的表示论，更好地称为表示论，致力于研究模。这些数学结构概括了向量空间的概念，并提供了一种媒介，在这种媒介中，乘法和加法的实例可以表现出来。模块的模型论主张从数理逻辑的角度来理解模块。因此，人们不仅对解线性方程组感兴趣，而且还对决定当一个句子在适当的形式语言中被赋予一个给定的解释时，它何时为真感兴趣。从这个观点最自然地产生的模块（也是这个项目的主题）是纯内射的，或代数紧致的模块。它们也显著地出现在表示论的经典（非逻辑）方法中，从而为代数和数理逻辑之间的进一步相互作用提供了有利的基础。

The theory of purity that arises in abelian group theory and, more generally,module theory may be developed (Crawley-Boevey) in the setting of a locally finitelypresented category. The prototypical example of such a category is the category offlat (left) modules over a ring R. The pure-injective objects of the category offlat R-modules are the flat cotorsion modules. The existence of cotorsion envelopesin the category of left R-modules is a consequence of the Flat Cover Conjecture(proved by Bican, El Bashir and Enochs). The project proposes to develop a theory offlat cotorsion modules that generalizes the classical theory of pure-injective modulesover R.Ring Theory is the area of Algebra that concentrates on the interaction between possiblenotions of multiplication and addition. The representation theory of rings, better knownas Representation Theory, is devoted to the study of modules. These mathematical structuresgeneralize the concept of a vector space and provide a medium where such instances ofmultiplication and addition may manifest themselves. The Model Theory of Modules proposesto understand modules from the point of view of Mathematical Logic. Thus one is not onlyinterested in solving linear equations, but also in deciding when a sentence in the appropriate formal language becomes true when it is assigned a given interpretation.The modules that arise most naturally from this point of view (and are the subject ofthis project) are the pure-injective, or algebraically compact, modules. They also occurprominently in the classical (non-logical) approach to Representation Theory and so providefruitful ground for further interaction between Algebra and Mathematical Logic.