In this paper we are interested in Hochschild cohomology of finite-dimensional algebras; the main motivation is to generalize group cohomology to larger classes of algebras. If suitable finite generation holds, one can define support varieties of modules as introduced by [SS]. Furthermore, when the algebra is self-injective, many of the properties of group representations generalize to this set...

We construct tame differential calculi coming from toral actions on a class of $\mathrm{C}^*$-algebras. Relying the existence unique Levi-Civita connection such calculus, we prove version Bianchi identity. A Gauss-Bonnet theorem for canonical calculus rank two is studied.