Exercise 3.17

Proof. $C$ is a flat $A$-module, and a faithfully flat $B$-module.

Let M, N be $A$-modules, $\varphi :M\to N$ be an injective map. Then, as $C$ is flat $A$-module, $\varphi {\text{⊗}}_{A}1:M{\text{⊗}}_{A}C\to N{\text{⊗}}_{A}C$ is injective (AM proposition 2.19). As $C\simeq C{\text{⊗}}_{B}B$, this means that $\varphi {\text{⊗}}_A(1{\text{⊗}}_{B}1):M{\text{⊗}}_A(B{\text{⊗}}_{B}C)\to N{\text{⊗}}_A(B{\text{⊗}}_{B}C)$ is injective. As tensor product commutes, $(\varphi {\text{⊗}}_{A}1){\text{⊗}}_{B}1:(M{\text{⊗}}_{A}B){\text{⊗}}_B\to (N{\text{⊗}}_{A}B){\text{⊗}}_{B}C$ is injective. As $C$ is faithfully flat $B$-module, $x\mapsto 1{\text{⊗}}_{B}x$ is injective, so $\varphi {\text{⊗}}_{A}1:M{\text{⊗}}_{A}B\to N{\text{⊗}}_{A}B$ is injective. Thus, B is a flat $A$-module, again by the proposition 2.19 □