  Let k be a commutative algebra with the field of the rational numbers
included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed
complex, simpler than the canonical one, giving the Hochschild and cyclic
homologies of E relative to ker(p). This complex resembles the canonical
reduced mixed complex of an augmented algebra. We begin the study of our
complex showing that it has a harmonic decomposition like to the one considered
by Cuntz and Quillen for the normalized mixed complex of an algebra.
