⊥N AS AN ABSTRACT ELEMENTARY CLASS
⊥N AS AN ABSTRACT ELEMENTARY CLASS
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⊥N 作为抽象小学课程
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
P. Eklof
中科院分区：
文献类型：
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作者：
J. Baldwin;P. Eklof
我们证明了抽象初等类（AEC）的概念为模块类的几个性质提供了一个统一的概念，并讨论了这些抽象初等类的稳定性。一个抽象的初等类由一组模型K和一个子模型<s:1> K的概念的强化组成，使得（K, <e:1> K）满足下面描述的性质。这里我们处理不同的类（N, <s:1> N）；下面给出了模⊥N的精确定义。一个关键的创新之处在于，A∈NB意味着A⊥B和B/A∈N。我们在文中定义了这里使用的主要概念；模块理论的重要背景定义和证明见[EM02]和[GT06]；AEC的概念源于Shelah（例如[She87]），但在[Bal]中收集。令人惊讶的事实是，类（N, <s:1> N）的一些基本模型理论性质直接转化为类⊥N和环R上的模N的代数性质，这些性质以前已经在完全不同的背景下研究过（模的近似理论，无限维倾斜理论等）。在环R上的强条件增加的情况下，我们的主要结果是：
We show that the concept of an Abstract Elementary Class (AEC) provides a unifying notion for several properties of classes of modules and discuss the stability class of these AEC. An abstract elementary class consists of a class of models K and a strengthening of the notion of submodel ≺K such that (K,≺K) satisfies the properties described below. Here we deal with various classes (N,≺N ); the precise definition of the class of modules ⊥N is given below. A key innovation is that A≺NB means A ⊆ B and B/A ∈ ⊥N . We define in the text the main notions used here; important background definitions and proofs from the theory of modules can be found in [EM02] and [GT06]; concepts of AEC are due to Shelah (e.g. [She87]) but collected in [Bal]. The surprising fact is that some of the basic model theoretic properties of the class (N,≺N ) translate directly to algebraic properties of the class ⊥N and the module N over the ring R that have previously been studied in quite a different context (approximation theory of modules, infinite dimensional tilting theory etc.). Our main results, stated with increasing strong conditions on the ring R, are: