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 Hochschild homology H-.(A,A) is isomorphic to the de Rham differential forms over 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 formula 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 functions 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 De Wilde and Lecompte. I will finally show how this formula can be used to construct an homotopy for the cyclic homology. References: 19
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