Exercise 8, § 3

Let A be a ring, B an A-algebra (not necessarily commutative) and E a finitely generated left B-module.

(a) Let S be a multiplicative subset of A and let $S^{-1}B = S^{-1}A \otimes_A B$ be given its $S^{-1}A$-algebra structure; $S^{-1}E$ can then be considered as a left $S^{-1}B$-module (Algebra, Chapter VIII, , no. 1) isomorphic to $S^{-1}B \otimes_B E$. Show that, if E is a flat B-module, $S^{-1}E$ is a flat $S^{-1}B$-module

(b) Let $(S_\lambda)_{\lambda \in L}$ be a family of multiplicative subsets of A such that, for every maximal ideal m of A, there exists $\lambda \in L$ such that $m \cap S_\lambda = \varnothing$. Show that $\bigoplus_{\lambda \in L} S_\lambda^{-1}B$ is a faithfully flat B-module (left or right; cf. § 2, Exercise 8). Show that, if, for all $A \in L$, $S_\lambda^{-1}E$ is a flat (resp. faithfully flat) $S_\lambda^{-1}B$-module, then E is a flat (resp. faithfully flat) B-module (use Chapter I, § 3, Exercise 9, with $C_\lambda = S_\lambda^{-1}A$).

(c) Suppose that the $S_\lambda$ satisfy condition (b) and that L is finite. Show that, if, for every $A \in L$, $S_\lambda^{-1}E$ is a finitely generated projective $S_\lambda^{-1}B$-module, then E is a finitely generated projective B-module.