Exact Categories
Exact Categories
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DOI：
10.1016/j.exmath.2009.04.004
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发表时间：
2008-11
期刊：
影响因子：
6.8
通讯作者：
Theo Buehler;Theo B Uhler
中科院分区：
文献类型：
--
作者：
Theo Buehler;Theo B Uhler
我们调查的基础上同调代数的准确范畴的意义上的奎伦。所有的图引理都是直接从公理证明的，特别是five引理，3×3引理和snake引理。简要讨论了正合函子、幂等完备性和弱幂等完备性。然后，我们表明，它是可能的，以建设派生类别的确切类别没有任何嵌入到阿贝尔类别，我们素描德利涅的方法派生函子。具有阿贝尔范畴中的值的经典导函子的构造可以轻松地转换为精确范畴，即，给出了射影分解的比较定理和马蹄引理的证明。在讨论了一些例子后，我们在附录中详细说明了Gabriel-Quillen嵌入定理的证明。
We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel–Quillen embedding theorem in an appendix.