Rings with several objects
Rings with several objects
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DOI：
10.1016/0001-8708(72)90002-3
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发表时间：
1972-02
影响因子：
1.7
通讯作者：
B. Mitchell
中科院分区：
文献类型：
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作者：
B. Mitchell
本文的目的是指出，大多数非交换同调环理论推广到（前）可加范畴理论，以及提供一些理由，希望它这样做。其中第一个暗示的可能性，这种推广来在弗洛伊德的论文，其中环理论的概念内射信封是联系在一起的阿贝尔范畴概念的正合性的函子，以产生弗洛伊德的显着证明阿贝尔群值嵌入定理的小阿贝尔范畴。从那时起，已经有几篇论文涉及取代定理环定理添加剂范畴。人们似乎没有普遍认识到的是，只要人们承认加法范畴的概念是环范畴的概念的完全自然的推广，那么程序就可以执行到什么程度的完备性。事实上，如果说某些环论事实能推广到如此遥远的环境最初让人感到惊讶，那么现在更令人惊讶的是，它们没有推广。历史上，加法范畴是从一些更难处理的例子中抽象出来的，比如给定环上所有模的范畴，而不是环本身的好例子。事实上，在关于这个主题的早期一篇更重要的论文中，它说：“虽然构造不能（有有限乘积）的预可加范畴很容易，但这样的例子似乎都是人为的，表明预可加范畴的概念在很大程度上可以被绕过。这句话不应该被解释为轻微反对环，而只是作为当时流行的信念，任何体面的性质，唯一的范畴必须是阿贝尔，或至少接近它的指示。在本文件的方法将是模仿尽可能密切的环理论的经典证明。虽然有时由于某些现象或其他现象可能在一个对象的情况下崩溃的事实，可能需要一些新的东西，但航行相对平稳，主要归功于添加剂Yoneda引理，在一个对象的情况下，这只是熟悉的自然同构
The purpose of this paper is to indicate that most of noncommutative homological ring theory generalizes to (pre) additive category theory, as well as to provide some reasons for wanting it to do so. One of the first hints of the possibility of such generalization came in Freyd’s thesis, where the ring theoretic notion of injective envelope was linked up with the abelian categorical notion of exactness of a functor to yield Freyd’s striking proof of the abelian group valued imbedding theorem for small abelian categories. Since then there have been several papers concerned with replacing theorems about rings by theorems about additive categories. What does not seem to be generally realized is the degree of completeness to which the program can be carried out, providing that one recognizes the notion of an additive category as an entirely natural generalization of that of a ring. Indeed, if it came initially as a surprise that certain ring theoretic facts generalize to such an apparently far-out setting, it now comes as even more of a surprise when they don’t.Historically, additive categories were abstracted from some of the more unwieldy examples, such as the category of all modules over a given ring, rather than the well established example of the ring itself. In fact, in one of the more important early papers on the subject, it is stated,“While it is easy enough to construct preadditive categories which fail to (have finite products), such examples all seem sufficiently artificial to suggest that the notion of a preadditive category can for the most part be by-passed”. This remark should not be interpreted as a slight against rings, but rather only as indicative of the belief prevalent at that time that the only categories with any decent properties must necessarily be abelian, or at least close to it. The approach in the present paper will be to imitate as closely as possible the classical proofs of ring theory. Although something new may be called for on occasion owing to the fact that some phenomenon or other may have collapsed in the one object case, the sailing is relatively smooth, thanks mainly to the additive Yoneda lemma, which in the one object case is nothing but the familiar natural isomorphism