Derived categories and stable equivalence
Derived categories and stable equivalence
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DOI：
10.1016/0022-4049(89)90081-9
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发表时间：
1989-11
影响因子：
0.8
通讯作者：
J. Rickard
中科院分区：
文献类型：
--
作者：
J. Rickard
Happel [6]和Cline，Parshall和Scott [4]证明了Happel和Ringel [8]的倾斜函子可以用所涉及的模范畴的导出范畴的等价来解释。在[lo]中，我们推广了这个结果，给出了这样一个等价的充分必要条件;在这种更一般的情况下，倾斜模必须用“倾斜复形”来代替，倾斜复形是满足类似于倾斜模所满足的条件的投射模的链复形（见下面的定理1.1）。本文的目的是表明，这种推广有有趣的应用程序，不出现更严格的倾斜模块。特别地，如果/I是一个自内射代数（例如，一个有限群的模群代数），那么很容易看出任何关于/1的倾斜模都是投射的-事实上，任何有限投射维数的/l-模都是投射的。因此，在这种情况下，经典的倾斜理论简化为森田等价。在更一般的情况下，倾斜复形，我们将表明，有许多应用的理论，以自我内射代数和第2节中，我们将表明，'派生等价'的自我内射代数是密切相关的稳定等价。已经有工作通过“平凡扩张代数”将倾斜理论和自内射代数联系起来;例如，Tachikawa和Wakamatsu[11]表明，如果r是从/1倾斜的有限维代数，则平凡扩张代数T/1和TT是稳定等价的。在第3节中，我们推广了他们的结果，并证明了它在导出等价性方面有一个非常自然的证明。事实上，一个倾斜复杂的A与自同态环rgives上升，通过张量与T/1，倾斜复杂的TA与自同态环TT我们希望派生的等价可能有有用的应用，模表示理论和第4节中，我们开始与最简单的情况下，块循环亏损群，并表明，布劳尔树代数是稳定的等价实际上是派生等价。
Happel [6] and Cline, Parshall and Scott [4] showed that the tilting functors of Happel and Ringel [8] can be interpreted in terms of an equivalence of derived categories of the module categories involved. In [lo] we generalised this result to give necessary and sufficient conditions for such an equivalence; in this more general case tilting modules must be replaced by ‘tilting complexes’, which are chain complexes of projective modules that satisfy conditions analogous to those satisfied by tilting modules (see Theorem 1.1 below). The aim of this paper is to show that this generalisation has interesting applications that do not arise for the more restrictive tilting modules. In particular, if/I is a self-injective algebra (for example, a modular group algebra for a finite group) then it is easy to see that any tilting module for/1 is projective-in fact, any/l-module of finite projective dimension is projective. Therefore in this case classical tilting theory reduces to Morita equivalence. In the more general case of tilting complexes we shall show that there are many applications of the theory to self-injective algebras and in Section 2 we shall show that ‘derived equivalence’for self-injective algebras is closely connected with stable equivalence. There has been work connecting tilting theory and self-injective algebras via ‘trivial extension algebras’; for example, Tachikawa and Wakamatsu[ll] showed that if r is a finite-dimensional algebra that is tilted from/1, then the trivial extension algebras T/1 and TTare stably equivalent. In Section 3 we generalise their result and show that it has a very natural proof in terms of derived equivalence. In fact a tilting complex for A with endomorphism ring rgives rise, by tensoring with T/1, to a tilting complex for TA with endomorphism ring TT We hope that derived equivalence may have useful applications to modular representation theory and in Section 4 we start with the simplest case, blocks with cyclic defect group, and show that the Brauer tree algebras that are stably equivalent are in fact derived equivalent.