The topic of this talk is an algebraic structure whose best known example is provided by the multivector fields and the differential forms on a smooth manfiold: the multivector fields are a Gerstenhaber algebra with respect to wedge product and Schouten-Nijenhuis bracket, and the differential forms are what we call a Batalin-Vilkovisky module over this Gerstenhaber algebra, which means that the multivector fields act in two ways on forms - by means of contraction and of Lie derivative - and that these actions are related by a differential that fits into Cartan's "magic" homotopy formula. Nest, Tamarkin and Tsygan suggested to refer to this abstract package of two graded vector spaces with such operations as to a noncommutative differential calculus.

Generalising work by the above mentioned authors and by Getzler, Gerstenhaber, Goodwillie, Huebschmann, Rinehart and others, I will explain that Ext and Tor over Hopf algebroids tends to carry such a structure which means that homological algebra produces plenty of examples of noncommutative differential calculi, including for example Hochschild and Poisson (co)homology.

As a first application, Ginzburg's theorem that the Hochshcild cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra is extended to twisted Calabi-Yau algebras such as quanum groups, quantum homogeneous spaces or quantum vector spaces. (Joint work with Niels Kowalzig)

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