manuscipta math. 77, 63 - 70 (1992) manuscripta mathemati ca ~) Springer-Verlag 1992 A Note on the Hochschild Homology and Cyclic Homology of a topological Algebra REINHOLD HUBL In this note we use a topological version of Hochschild homology and cyclic homology of a commutative algebra, introduced by P. Seibt in [Se2], to show, that periodic homology can be used to calculate the relative algebraic de Rham cohomology of a morphism of affine Q-schemes of finite type as defined in [Ha], chapt. III, w Cyclic homology, introduced in [Co], [Ts], has been studied in a series of papers as a non-commutative generalization of de Rham cohomology (c.f. [Col, [LQI , [FT], tSell, XI.1). Xn fact by [LQ], (2.9) cyclic cohomology and periodic cohomology can be used to calculate the relative algebraic de Rham cohomology of a smooth morphism of affine Q-schemes. In [Se2] P. Seibt introduced a local version of cyclic and Hochschild homology (see also [Hii], w and in this note we shall use this theory and some commutative algebra to calculate the relative algebraic de Rham cohomology of any morphism of affine Q-schemes of finite type as defined in [Ha], chapt. III, w generalizing a result of Feigin and Tsygan. Let k be a unital commutative ring and let R be a unital associative k-algebra. A fundamental system of rteighbourhoods of 0 is a family 11 of two sided ideals of R such that for any two I, J 6 11 there exists a K 6 11 with K _C I f3 J. Such a it defines a linear topology r on R, and we call (R,r)/k a topological algebra. By (C,(R/k), b, B) we denote the Hochschild complex of R/k ( considered as a 63