Let k be the field C or R, let M be the space k(n) nod let A be the algebra
of polynomials over M. We produce a general explicit homotopy formula betw
een the complex of Hochschild chains of A, (C.(A), b) and the de Rhan compl
ex of M, (Omega. (M), 0). This formula can be generalized when M is a contr
actible open set in a complex manifold and A is the space of holomorphic fu
nctions over M. Then we find a new, homotopy formula for rite Hochschild co
homology of the algebra of smooth fonctions over M different from the one g
iven in [5]. Finally, we show how this formula carl be used to construct a
homotopy for the cyclic homology. (C) 1999 Academie des Sciences/Editions s
cientifiques et medicales Elsevier SAS.