Exercise 31, § 2

a) Let $A$ be a noetherian ring, $m$ a maximal ideal of $A$, $p$ a prime ideal contained in $m$; now prove that if the separated completion $\hat{A}$ of $A$ for the $m$-adic topology is integral, the filter basis of the symbolic powers $p^{(n)}$ of $p$ tends to $0$ for the $m$-adic topology in $A$. (Reduce to the case where $A$ is local; let $q$ be a prime ideal of $\hat{A}$ whose trace on $A$ is $p$, and for every $n > 0$, let $c_n \notin p$ such that $c_n p^{(n)} \subset p^n$; now prove that one has $c_n p^{(n)} \hat{A} \subset q^{(n)}$. Use Exercise 29 e), as well as Chapter III, § 2, No. 7, Proposition 8.)

b) Let $A$ be a noetherian ring, $n$ an ideal of $A$, $p$ a minimal prime ideal in $\operatorname{Ass}(A/n)$, $m_0, m$ two maximal ideals of $A$ containing $p$; denote by $A(n), A(m_0), A(m)$ the separated completions of $A$ for the $n$-adic, $m_0$-adic, $m$-adic topologies respectively. Let $z$ be an element of $A(n)$ whose canonical image in $A(m_0)$ is zero. Now prove that if $A(m)$ is integral, the canonical image of $z$ in $A(m)$ is zero also. (Take $z$ as the limit of a sequence $(x_n)$ of elements of $A$ such that $x_n - x_m \in n^m$ for $n \geq m$, and $x_n \in m_0^n$; deduce that $x_m$ belongs to the closure of $n^m$ for the $m_0$-adic topology; using Exercise 8, conclude that $x_m \in p^{(m)}$ for every $m$; complete using a.)

c) Let $A$ be a Zariski ring, $r$ a defining ideal of $A$; suppose that $\operatorname{Spec}(A)$ is connected and that for every maximal ideal $m$ of $A$, the separated completion $A(m)$ of $A$ for the $m$-adic topology is integral. Now prove that the completion $\hat{A}$ of $A$ for the $r$-adic topology is integral. (Now prove that for every maximal ideal $m$ of $A$, the canonical homomorphism $\hat{A} \to A(m)$ is injective. For this, one may suppose that $r$ is the intersection of a finite number of prime ideals, minimal in $\operatorname{Ass}(A/r)$, namely $r = p_1 \cap p_2 \cap \ldots \cap p_s$; now prove that the assumption on $\operatorname{Spec}(A)$ implies for each $i$ the existence of a maximal ideal $m_i$ containing $(p_1 \cap \ldots \cap p_{i-1}) + p_i$. Using $b$, now prove that if the canonical image of $z \in \hat{A}$ in $A(m)$ is zero and if, for example, $p_1 \subset m$, then the canonical image of $z$ in each of the $A(m_i)$ is zero, then conclude that the canonical image of $z$ in $A(m')$ is zero for every maximal ideal $m'$ of $A$.)