导出范畴与投射类
Derived Categories and Projective Classes
摘要：
模型范畴的一个重要例子是R-模的无界链复形范畴，它的同伦范畴是环R的导出范畴。这个例子表明传统的同调代数被奎伦同调代数所包含。本文的目的是表明，更一般形式的同调代数也适合奎伦的框架。具体地说，完备且上完备的阿贝尔范畴A中的任何对象集都生成A上的一个投射类，这正是在A中做同调代数所需的信息。主要结果是：如果生成对象在适当意义上是“小”的，则A的对象的链复形范畴具有反映投射类的同调代数的模型范畴结构.本文的动机是构造环R的“纯导出范畴”。最后，我们解释了如何在一个可能的非阿贝尔范畴的单纯对象的类别可以配备一个模型类别结构，反映了一个给定的投影类。
英文摘要：
An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, any set of objects in a complete and cocomplete abelian category A generates a projective class on A, which is exactly the information needed to do homological algebra in A. The main result is that if the generating objects are ``small'' in an appropriate sense, then the category of chain complexes of objects of A has a model category structure which reflects the homological algebra of the projective class. The motivation for this work is the construction of the ``pure derived category'' of a ring R. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class.
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