Proposition 2

Let R, S be two rings, $\rho : \mathbf{R} \to \mathbf{S}$ a homomorphism and F a left R-module. The following two properties are equivalent: (a) $\operatorname{Tor}_1^R(\rho_*(E), F) = 0$ for every right S-module E. (b) The left S-module $\varphi^*(F) = F_{(S)} = S \otimes_R F$ is flat and $\operatorname{Tor}_1^R(\rho_*(S_d), F) = 0$.

Suppose that (a) holds. Taking $E = S_d$, we see that $\operatorname{Tor}_1^R(\rho_*(S_d), F) = 0$. We show also that $F_{(S)}$ is a flat S-module. For that, we note that, if E is a right S-module, the additive group $E \otimes_S F_{,,}$ is identified with $\rho_*(E) \otimes_R F$. Then if there is an exact sequence of right S-modules

we obtain, using (a), an exact sequence

or also

which proves that $F_{(S)}$ is flat.

Conversely, if (b) holds, we have first of all, for every free right S-module $L = S_d^{(I)}$, $\operatorname{Tor}_1^R(\rho_*(L), F) = (\operatorname{Tor}_1^R(\rho_*(S_d), F))^{(I)} = 0$. Every right S-module E can be written in the form $E = L/H$ for a suitable free S-module L; then we have the exact sequence

(2) $0 = \operatorname{Tor}_1^R(\rho_*(L), F) \to \operatorname{Tor}_1^R(\rho_*(E), F) \to \rho_*(H) \otimes_R F \to \rho_*(L) \otimes_R F.$

But as $F_{,,}$ is flat, the homomorphism $H \otimes_S F_{(S)} \to L \otimes_S F_{(S)}$ is injective and is identified with the homomorphism

Then it follows from (2) that $\operatorname{Tor}_1^R(\rho_*(E), F) = 0$.