Derived categories and Morita theory
Derived categories and Morita theory
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DOI：
10.1016/0021-8693(86)90224-3
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发表时间：
1986-12
影响因子：
0.9
通讯作者：
E. Cline;B. Parshall;L. Scott
中科院分区：
文献类型：
--
作者：
E. Cline;B. Parshall;L. Scott
本文类似于环的模范畴的经典理论[L，Lo]，首次尝试构造了派生范畴的Morita理论。我们的动机来自于导出范畴在李代数和代数群的表示理论中的最近重要性[3,111。然而，我们在这里的观点完全是环论，目的是将最近由Brenner和Butler[S]、Bongartz[4]、Happl和Ringel[S]以及Happl[9]发展的有限维代数的倾斜模理论置于更广泛的背景下。在第一节中，我们简要回顾了三角范畴理论中的一些标准符号和结构。在第二节中，我们引入了环的广义倾斜模的概念，并证明了它是如何得到某些派生范畴的等价的。这一节的主要定理，即定理2.1，代表了Happl[9]的一个结果的推广和推广，尽管我们只是在深入研究了Happl和Bongartz的工作之后才得到它。在第三节中，我们证明了倾斜模条件在刻画派生范畴的某些等价时自然而必然地出现。最后，在第四节中，我们试图将迄今为止发展起来的森田理论放在派生范畴的一般森田理论的背景下，指出一些剩余的困难。在文[2]的精神下，我们还使用了部分建立的倾斜模块来获得本地化结果。
This paper represents a first attempt to construct a Morita theory for derived categories, analogous to the classical theory for module categories of rings [l, lo]. Our motivation comes from the recent importance of derived categories in the representation theory of Lie algebras and algebraic groups [3, 111. However, our point of view here is entirely ringtheoretic, aimed at placing into a broader context the recent theory of tilting modules for finite dimensional algebras as developed by Brenner and Butler [S], Bongartz [4], Happel and Ringel [S], and Happel [9]. In Section 1 we briefly recall some standard notation and constructions from the theory of triangulated categories. In Section 2 we introduce the notion of a generalized tilting module for rings and show how it gives rise to equivalences of certain derived categories. The main theorem of this section, Theorem 2.1, represents an extension and generalization of a result a Happel [9], although we obtained it only after thorough study of Happel’s work as well as that of Bongartz [4]. In Section 3 we show that the tilting module conditions arise naturally and necessarily in characterizing certain equivalences of derived categories. Finally in Section 4 we try to place the Morita theory developed so far in the context of a general Morita theory for derived categories, indicating some of the remaining difficulties. We also use part of the tilting module set up to obtain a localization result in the spirit of [2].