Homologies of cubic Artin–Schelter regular algebras

The Hochschild homology of cubic Artin–Schelter regular algebras of type A with generic coefficients is computed. We follow the method used by van den Bergh [K-Theory 8 (1994) 213–230] in the quadratic case and consider these algebras as deformations of a polynomial algebra with remarkable Poisson brackets. A new morphism of resolutions is introduced. The de Rham cohomology, cyclic and periodic cyclic homologies, and the Hochschild cohomology are also computed.