Exercise 14, § 1

Let $A$ be a complete noetherian local ring.

a) Exercise prove that an ideal of $A$ which is contained in the union of a sequence of prime ideals of $A$ is contained in one of them (observe that the topological space $A$ is a Baire space, cf. TG, IX, p. 55, Theorem 1).

b) Let $M$ be an $A$-module admitting a countable generating family. Exercise prove that the conclusions of th. 2 are still satisfied (observe that the set $\mathrm{Ass}(M)$ is countable, and deduce from a) that if $\mathrm{prof}_A(J; M) > 0$, there exists an element $x$ of $J$ such that the homothety $x_M$ is injective).