Applications of model theory to cotorsion modules

在阿贝尔群理论和更一般的模理论中产生的纯洁性理论可以在局部有限表示范畴的背景下得到发展(Crawley-Boevey)。这类范畴的典型例子是环R上的左(左)模范畴。环R-模范畴的纯内射对象是平坦上扭模。左R-模范畴中余挠包络的存在是平盖猜想的结果(由Bcan，El Bashir和Enochs证明)。该项目建议发展一种关于Lat余扭模的理论，它推广了R上的纯内射模的经典理论。环理论是代数的一个领域，专注于乘法和加法的可能运动之间的相互作用。环的表示论，即众所周知的表示论，致力于模的研究。这些数学结构推广了向量空间的概念，并提供了一种媒介，在这里乘法和加法的实例可以表现出来。模块模型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，而且还对确定当一个适当的形式语言中的句子被赋予给定的解释时何时成为真。从这个角度最自然地产生的模块(也是本项目的主题)是纯内射的，或者说是代数紧的。它们也突出地出现在表示论的经典(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可靠的基础。

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.