A condition in constructing chain homotopies (Q1081890)

On the suggestion of M. Morimoto the author asks whether or not the statement below is true depending on the ring K and the group G. Statement: Let \(R=K[G]\). If \(f_*,g_*: C_*\to D_*\) are chain equivalences of two free R-chain complexes, inducing \(f_*=g_*: H(C_*)\to H(D_*)\) then \(f_.\) is chain homotopic to \(g_.\); \(f_.\simeq g_.\). The main theorem is: a) For any group G and \(R={\mathbb{Z}}[G]\) the statement does not hold. b) If K is a field, G a finite group the statement holds if and only if (ch K,\(| G|)=1\) (including the case ch K\(=0)\). c) When K is a ring, G a group which contains an element of infinite order and \(R=K(G)\), then the statement does not hold. - The problem arises in context with papers (preprints) of K. H. Dovermann, M. Rothenberg and of M. Morimoto.