Exercise 2, § 4

Let $p:A\to B$ be a homomorphism of rings, M an A-module. Now prove that the following properties are equivalent:

α) One has $\operatorname{Tor}^A_1(p_*(P),M)=0$ for every B-module on the right P.

EXERCISES

β) One has Tor^A_1(ρ_*(P), M) = 0 for every monogenic B-module on the right P. γ) The B-module B ⊗_A M is flat, and one has Tor^A_1(B, M) = 0. (To see that γ) implies α), consider an exact sequence 0 → R → L → P → 0 of B-modules on the right, where L is free.)