A Condition In Constructing Chain Homotopies

In this note we show by an example that the answer is negative, namely that f. —g, does not always hold. We also consider the case R = K[G^ where K is a field, and show that the answer is negative if and only if the characteristic of the field K divides the order of G when G is a finite group. We also obtain some condition for an infinite group G. If 1. 1 were true, then the arguments of Morimoto in [3], which computes a homology class of the torsion invariant (cf, Theorem 8. 4 of Dovermann-Rothenberg [2]), would be considerably simplified. Thus the negativeness of 1. 1 suggests that we cannot do without delicate arguments as in [3] in computing torsion invariants. The author would like to thank M. Morimoto, T. Matumoto and K. Motose for valuable conversations. He would also like to thank