Return to Question

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids?

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids?

Clarification: It is known that 1-dimensional Hochschild cohomology of the Group algebra C[G] is isomorphic to so-called external derivations of C[G]. On the other hand, it is known that the space of derivations can be identified with so-called characters on a groupoid aG of adjoint actions of the group G.

Therefore 1-dimensional Hochschild cohomology can be identified with 1-dimensional cohomology of the Cayley complex of the groupoid aG in the case when the group G is a finite presented group. I would like to know if a similar identification is known for the Hochschild cohomologies of higher dimensions

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids? For example, if $G$ is a group, is there a natural $G$-module $M$ such that the Hochschild cohomology of $\mathbb{Z}[G]$ is naturally isomorphic to the group cohomology of $G$ with coefficients in $M$?

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids?

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groups? For example, if $G$ is a group, is there a natural $G$-module $M$ such that the Hochschild cohomology of $\mathbb{Z}[G]$ is naturally isomorphic to the group cohomology of $G$ with coefficients in $M$?

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids? For example, if $G$ is a group, is there a natural $G$-module $M$ such that the Hochschild cohomology of $\mathbb{Z}[G]$ is naturally isomorphic to the group cohomology of $G$ with coefficients in $M$?