On the cohomology groups of an associative algebra
On the cohomology groups of an associative algebra
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DOI：
10.2307/1969145
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发表时间：
1945
影响因子：
4.9
通讯作者：
G. Hochschild
中科院分区：
文献类型：
--
作者：
G. Hochschild
结合代数的上同调理论研究了代数W到双边W-模A的m-线性映射。在这一理论中，2 I到$的m-线性映射的加法群（2（m）：$）在组合拓扑中起着类似于m维上链群的作用。Eilenberg和MacLane定义了（2[（m）：93）到（2（m+1）：3）的线性映射，类似于组合拓扑的上边界算子，并引出了$“上同调群”的概念。一维和二维的特殊情况（S到一个双边21-模的线性和双线性映射）以前已经出现过，与Whitehead的第一和第二引理有关。在某种意义上，结合代数的上同调理论是退化的：1维上同调群已经决定了所有其他群。事实上，如果$是任意的双边Et-模，我们可以构造另一个双边S-模，（21：3），使得（当m > 2时）2 t的m维$-上同调群与2的（m l）维（21：$）-上同调群同构[（定理3.1）]。本文主要研究代数的结构与其上同调群的消失之间的关系。证明了一个代数是可分的当且仅当它的所有上同调群为零（定理4.1）。这是对以前非模基场3的结果的推广。一个代数的二维上同调群直接与A的“扩张”相连，即代数e3，其中21是同态像。特别地，21的所有2维上同调群为零的条件表示2f的每个扩张具有e= * + S的形式，其中2[*]是与21同构的子代数，S是e3到51的同态的核（定理6.1）。这是连接与（广义）第三结构定理Wedderburn这可能是说，即二维上同调群的可分代数消失。更一般地，人们会对条件Cm的结构意义感兴趣：“所有m维上同调群消失。“定理3.1意味着C.+，是Cm的一个结果，m > 1。但这是一个悬而未决的问题，是否Cm和C.+，是等价的。
The cohomology theory of associative algebras is concerned with the m-linear mappings of an algebra W into a two-sided W-module A. In this theory, the additive group (2(m):$) of the m-linear mappings of 2I into $ plays a r6le analogous to that of the group of m-dimensional cochains in combinatorial topology. A linear mapping of (2[(m):93) into (2(m+1):3) analogous to the coboundary operator of combinatorial topology and leading to the notion of $"cohomology group" has been defined by Eilenberg and MacLane'. The special cases of dimension one and two (linear and bilinear mappings of S into a two-sided 21-module) have appeared before in connection with the so-called first and second lemmas of Whitehead2. In a sense, the cohomology theory of associative algebras is degenerate: the 1-dimensional cohomology groups already determine all the others. In fact, if $ is any two-sided Et-module, one can construct another two-sided S-module, (21:3), such that (for m > 2) the m-dimensional $-cohomology group of 2t is isomorphic with the (m l)-dimensional (21:$)-cohomology group of 2[ (Theorem 3.1). The present paper is concerned primarily with the connections between the structure of an algebra and the vanishing of its cohomology groups. It is shown that an algebra is separable if and only if all its cohomology groups vanish (Theorem 4.1). This is a generalization of a result obtained previously for the case of a non-modular ground field3. The 2-dimensional cohomology groups of an algebra are directly connected with the "extensions" of A, i.e. algebras e3 of which 21 is a homomorphic image. In particular, the condition that all 2-dimensional cohomology groups of 21 vanish signifies that every extension of 2f has the form e= * + S, where 2[* is a subalgebra isomorphic with 21 and S is the kernel of the homomorphism of e3 onto 51 (Theorem 6.1). This is connected with the (generalized) third structure theorem of Wedderburn which may be stated by saying, that the 2-dimensional cohomology groups of a separable algebra vanish. More generally, one would be interested in the structural significance of the condition Cm: "all m-dimensional cohomology groups vanish." Theorem 3.1 implies that C.+, is a consequence of Cm, for m > 1. But it is an open question whether or not Cm and C.+, are equivalent.