Hochschild cohomology of finite—dimensional algebras
Hochschild cohomology of finite—dimensional algebras
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DOI：
10.1007/bfb0084073
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发表时间：
1989
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影响因子：
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通讯作者：
D. Happel
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文献类型：
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作者：
D. Happel
设k是代数闭域，A是有限维k-代数(结合的，带单位的)。我们用MODA表示有限生成的左A-模范畴。设AXA是有限生成的A-双模。Hochschild上同调群Hi(A，X)(I0)是由Hochschild[HO]引入的(定义见第1节)。低维群(I：：；2)对经典的代数结构有非常具体的解释，例如导子和扩张。格斯滕哈伯[GEJ]观察到，与代数几何也有联系。事实上，H2(A，A)控制着A的形变理论，并且证明了满足H2(A，A)=0的代数A是刚性的。对于类似的方法，我们还提到了加布里埃尔[GA L]的一篇文章。尽管如此，在具体的有限维代数类的实际计算中所做的工作却很少。在第一节中，我们简要回顾了Hochschild上同调的基本定义，并包括了人们经常用于直接计算的另一种描述。在第二节中，我们给出了一些计算。这包括关于CIBILS的结果的报告[ELJ，[C2J，[C3J和Gertenhaber和Schack[gsj.对于这些结果中的一些，我们已经包括了证明和一些例子。在第3节中，我们讨论派生。在剩下的两节中，我们概述了最近在有限维代数表示理论中出现的方法如何产生关于Hochschild上同调的信息。
For this let k be an algebraically closed field and A a finite-dimensional k-algebra (associative, with unit). By modA we denote the category of finitely generated left A-modules. Let AXA be a finitely generated A-bimodule. The Hochschild cohomology groups Hi (A, X)(i 0) were introduced by Hochschild [Ho](for a definition see section 1). The lowdimensional groups (i::; 2) have a very concrete interpretation of classical algebraic structures such as derivations and extensions. It was observed by Gerstenhaber [GeJ that there are also connections to algebraic geometry. In fact, H2 (A, A) controls the deformation theory of A. And it was shown that the algebras A which satisfy H2 (A, A)= 0 are rigid. For a similar approach we also mention an article of Gabriel [Ga L]. Despite this very little was done in actual computations for particular classes of finitedimensional algebras. In section 1 we briefly review the fundamental definitions of Hochschild cohomology and include an alternative description which one often uses for direct computations. In section 2 we present some computations. This includes a report on results due to Cibils [elJ,[C2J,[C3J and Gerstenhaber and Schack [GSJ. For some of these results we have included proofs and some examples. In section 3 we deal with derivations. In the remaining two sections we outline how recently emerged methods in the representation theory of finite-dimensional algebras yield information on the Hochschild cohomology.