Applications of model theory to cotorsion modules

在阿贝尔群论和更普遍的模块理论中出现的纯粹理论可以在局部有限呈现范畴的背景下发展（Crawley-Boevey）。这种范畴的典型例子是环r上的平面（左）模范畴。平面r模范畴的纯内射对象是平面扭转模。在左r模范畴中存在扭转包络是平盖猜想（由Bican， El Bashir和Enochs证明）的结果。该项目提出了一个关于平扭模的理论，它推广了经典的纯内射模理论。环理论是代数的一个领域，专注于乘法和加法的可能性之间的相互作用。环的表示理论，更广为人知的是表示理论，致力于研究模块。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。模块模型理论提出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感兴趣，而且还对决定一个用适当形式语言表达的句子在被赋予给定解释时何时为真感兴趣。从这个角度来看，最自然产生的模块（也是本项目的主题）是纯注入或代数紧化模块。它们也出现在表征理论的经典（非逻辑）方法中，因此为代数和数理逻辑之间的进一步互动提供了丰富的基础。

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.