Theorem 1

a) If one of the resolutions $R$ or $S$ is flat, then $\psi(S, R)$ is an isomorphism of graded $k$-modules.

b) If $R$ is projective or if $E$ is injective, $\varphi(R, E)$ is an isomorphism of graded $k$-modules.

a) Suppose for example $\mathbf R$ flat, and choose $\alpha$ and $\beta$ as above. The homomorphism $\beta\otimes\alpha$ is the composite of the morphisms

Since $L(P)$ (resp. $\mathbf R$) is flat and $\alpha$ (resp. $\beta$) is a homomorphism, $1_{L(P)}\otimes\alpha$ (resp. $\beta\otimes1_{\mathbf R}$) is one, by prop. 4 of X, p. 67. Therefore $\beta\otimes\alpha$ is a homomorphism and $\psi(S,R)=H(\beta\otimes\alpha)$ is bijective.

b) One reasons analogously, using prop. 4 of X, p. 86.