  Let k be a characteristic zero field, C a k-algebra and M a square zero two
sided ideal of C. We obtain a new mixed complex, simpler that the canonical
one, giving the Hochschild and cyclic homologies of C relative to M. 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. We also give new proofs of two theorems of Goodwillie,
obtaining an improvement of one of them.
