There are two approaches to the study of the cohomology of group algebras ℝ[G]: the Eilenberg-MacLane cohomology and the Hochschild cohomology. In the case of Eilenberg-MacLane cohomology, one has the classical cohomology of the classifying space BG. The Hochschild cohomology represents a more general construction, in which the so-called two-sided bimodules are considered. The Hochschild cohomology and the usual Eilenberg-MacLane cohomology are coordinated by moving from bimodules to left modules. For the Eilenberg-MacLane cohomology, in the case of a nontrivial action of the group G on the module M_l, no reasonable geometric interpretation has been known so far. The main result of this paper is devoted to an effective geometric description of the Hochschild cohomology. The key point for the new geometric description of the Hochschild cohomology is the new groupoid Gr associated with the adjoint action of the group G. The cohomology of the classifying space BGr of this groupoid with an appropriate condition for the finiteness of the support of cochains is isomorphic to the Hochschild cohomology of the algebra ℝ[G]. Hochschild homology is described in the form of homology groups of the space BGr, but without any conditions of finiteness on chains.

中文翻译：

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从几何角度看群代数的Hochschild同调和Eilenberg-MacLane同调之间的相关性

研究群代数ℝ[ G ]的同调性的方法有两种：Eilenberg-MacLane同调性和Hochschild同调性。对于Eilenberg-MacLane同调，具有分类空间BG的经典同调。Hochschild同调表示一种更一般的构造，其中考虑了所谓的双面双模。通过从双模块移动到左模块来协调Hochschild谐调和通常的Eilenberg-MacLane谐调。对于Eilenberg-MacLane同调，在基团G对模块M _l进行非平凡作用的情况下，到目前为止，尚无合理的几何解释。本文的主要结果致力于对Hochschild同调学进行有效的几何描述。Hochschild谐函数的新几何描述的关键点是与G组的伴随作用相关的新的类群Gr 。该类群的分类空间BGr的同构性具有适当的条件，以支持共链的有限性。代数ℝ[ G ]的Hochschild同调的同构。Hochschild同源性以空间BGr的同源性组的形式描述，但在链上没有任何限定性条件。