Exercise 19, § 3

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$ and $S$ a multiplicative subset of $A$. Let $A$ be a given the $m$-adic topology.

(a) Show that the ring $A\{S^{-1}\}$ ($\S$ 2, Exercise 27) is a flat $A$-module.

(b) Let $S'$ be another multiplicative subset of $A$ contained in $S$. Show that $A\{S^{-1}\}$ is a flat $(A\{{S'}^{-1}\})$-module (use Exercise 27 (d) of $\S$ 2).

(c) Show that $A\{S^{-1}\}$ is a flat $A_{(S)}$-module ($\S$ 2, Exercise 27 (g)).

(d) Suppose that $S = A - p$, where $p$ is an open prime ideal of $A$. Show that $A\{S^{-1}\}$ is a faithfully flat $A_{(S)}$-module and deduce that the ring $A_{,,}$ is Noetherian (cf. $\S$ 2, Exercise 27 (g)).