Homotopic formulas between Hochschild and De Rham complexes

Let k be the field C or R, let M be the space k(n) and let A be the algebra of polynomials over M. We know from Hochschild and co-workers that the Hoc hschild homology H-.(A,A) is isomorphic to the de Rham differential forms o ver M: this means that the complexes (C-.(A,A),b) and (Omega (.)(M), 0) are quasi-isomorphic. In this work, I produce a general explicit homotopy form ula between those two complexes. This formula can be generalized when M is an open set in a complex manifold and A is the space of holomorphic functio ns over M. Then, by taking the dual maps, I find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M (when M is either a complex or a real manifold) different from the one given by D e Wilde and Lecompte. I will finally show how this formula can be used to c onstruct an homotopy for the cyclic homology.