Exercise 7, § 5

Let $(A_\alpha, f_{\mu\lambda})$ be a directed direct system of Noetherian local rings; if $m_\lambda$ is the maximal ideal of $A_\lambda$, suppose that, for $\lambda \leq \mu, m_\mu = m_\lambda A_\mu$ and that $A_\mu$ is a flat $A_\lambda$-module. Show then that $A = \varprojlim A_\lambda$ is Noetherian and a flat $A_\lambda$-module for all $\lambda$. (If $m = m_\lambda A$ is the maximal ideal of $A$ (Chapter II, § 3, Exercise 16), show that on $A$ the $m$-adic topology is Hausdorff, observing that $A$ is a faithfully flat $A_\lambda$-module for all $A$. Then prove that, if $\hat{A}$ is the completion of $A$ with respect to the $m$-adic topology, $\hat{A}$ is Noetherian, using § 2, no. 10, Corollary 5 to Theorem 2; finally, prove that, for all $A$, $\hat{A}$ is a flat $A$-module using Theorem 1 of no. 2 and Proposition 2 of no. 3.)