Exercise 31, § 2

(a) Let A be a Noetherian ring, m a maximal ideal of A and p a prime ideal contained in m; show that, if the Hausdorff completion \hat{A} of A with respect to the m-adic topology is an integral domain, the filter base of symbolic power p^{(n)} of p tends to 0 with respect to the m-adic topology on A. (Reduce it to the case where A is local; let q be a prime ideal of $\hat{A}$ whose trace on A is p and for all $n > 0$ let $c_n \notin p$ be such that $c_n p^{(n)} \subset p^n$; show that $c_n p^{(n)} \hat{A} \subset q^{(n)}$. Use Exercise 29 (e) and 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 Ass(A/n) and $m_0, m$ two maximal ideals of A containing p; let A(n), A(m_0), A(m) denote the Hausdorff completions of A with respect to 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. Show that if A(m) is an integral domain, the canonical image of z in A(m) is also zero. (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 $n^m$ with respect to the $m_0$-adic topology; using Exercise 8, conclude that $x_m \in p^{(m)}$ for all m and complete with the aid of (a).)

(c) Let A be a Zariski ring and r a defining ideal of A; suppose that Spec(A) is connected and that for every maximal ideal m of A the Hausdorff completion A(m) of A with respect to the m-adic topology is an integral domain. Show then that the completion $\hat{A}$ of A with respect to the r-adic topology is an integral domain. (Show that for every maximal ideal m of A, the canonical homomorphism $\hat{A} + A(m)$ is injective. For this, we may assume that r is the intersection of a finite number of prime ideals which are minimal in Ass(A/r), let $r = p_1 \cap p_2 \cap \cdots \cap p_s$; show that the hypothesis on Spec(A) implies for each i the existence of a maximal ideal m, containing $(p_1 \cap \ldots \cap p_{i-1}) + p_i$. Using (b), show that, if the canonical image of $z \in \hat{A}$ in A(m) is zero and 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.)