Exercise 9, § 3

Let $A$ be a commutative ring, $B$ an $A$-algebra, $(C_\lambda)_{\lambda \in L}$ a family of $A$-algebras and $B_\lambda = C_\lambda \otimes_A B$ the tensor product algebra of $C_\lambda$ and $B$ for all $\lambda \in L$. Let $E$ be a left $B$-module. Set $E_\lambda = B_\lambda \otimes_B E = C_\lambda \otimes_A E$; this is a $(B_\lambda, C_\lambda)$-bimodule. Similarly, if $F$ is a right $B$-module, set

which is a $(C_\lambda, B_\lambda)$-bimodule.

(a) Show that the $(C_\lambda, C_\lambda)$-bimodule $F, \otimes_{B_\lambda} E_\lambda$ is isomorphic to $(F \otimes_B E) \otimes_A C_\lambda$.

(b) Show that, if $E$ is a flat (resp. faithfully flat) $B$-module, each of the $E_\lambda$ is a flat (resp. faithfully flat) $B_\lambda$-module. The converse is true if we assume further that $\bigoplus_{\lambda \in L} C_\lambda$ is a faithfully flat $A$-module.

(c) Show that, if $L$ is finite, each of the $E_\lambda$ is a finitely generated projective $B_\lambda$-module and $\bigoplus_{\lambda \in L} C_\lambda$ is a faithfully flat $A$-module, then $E$ is a finitely generated projective $B$-module (use Proposition 12 of no. 6).