Proposition 17

Let $A$ be a commutative ring and $(m_{\lambda})_{\lambda \in L}$ a family of ideals of $A$, distinct from $A$, such that $m_{\lambda}$ and $m_{\mu}$ are relatively prime for $\lambda \neq \mu$. For every family $s = (s(\lambda))_{\lambda \in L}$ of integers $\geq 0$, of finite support, set $a_s = \bigcap_{\lambda \in L} m_{\lambda}^{s(\lambda)}$ (equal to the product of the $m_{\lambda}^{s(\lambda)}$ for the $\lambda$ such that $s(\lambda) \neq 0$; cf. Chapter 11, § 1, no. 2, Propositions 3 and 5); the $a_s$ form a fundamental system of neighbourhoods of 0 with respect to a topology $\mathscr{T}$ compatible with the ring structure on $A$; let $\hat{A}$ be the Hausdorff completion of $A$ with respect to this topology. On the other hand, for all $\lambda \in L$, let $A_{\lambda}$ be the ring $A$ with the $m_{\lambda}$-adic topology and let $\hat{A}_{\lambda}$ be its Hausdorff completion. If $u : A \to \prod_{\lambda \in L} A_{\lambda}$ denotes the diagonal homomorphism, $u$ is continuous and the corresponding homomorphism $\hat{u} :$

(General Topology, Chapter III, § 6, no. 5 and Chapter II, § 3, no. 9, Corollary 2 to Proposition 18) is a topological ring isomorphism.

The first assertion follows from General Topology, Chapter III, § 6, no. 3, Example 3. Let us set $B = \prod_{\lambda \in L} A_{\lambda}$; as the topology on $A$ is finer than each of the $m_{\lambda}$-adic topologies, the mappings $\mathrm{pr}_{\lambda} \circ u$ are continuous and hence $u$ is continuous. Also, $u(a_s)$ is the intersection of the diagonal $A$ of $B$ and the open set $\bigcap_{\lambda \in L} \mathrm{pr}_{\lambda}^{-1}(m_{\lambda}^{s(\lambda)})$ of $B$; it follows that $u$ is a strict morphism from the additive group $A$ to $B$ with image $A$. Now $\Delta$ is dense in $B$. For let $b = (a_{\lambda})_{\lambda \in L}$ be an element of $B$; every neighbourhood of $b$ in $B$ contains a set of the form $b + V$, where $V = \bigcap_{\lambda \in L} \mathrm{pr}_{\lambda}^{-1}(m_{\lambda}^{s(\lambda)})$ for a family $s = (s(\lambda))_{\lambda \in L}$ with finite support of integers $\geq 0$. As the $m_{\lambda}^{s(\lambda)}$ are relatively prime in pairs (Chapter II, § 1, no. 2, Proposition 3), there exists $x \in A$ such that $x \equiv a_{\lambda} \pmod{m_{\lambda}^{s(\lambda)}}$ for all $\lambda$ (loc. cit., Proposition 5) and hence $(b + V) \cap A \neq \varnothing$. The Hausdorff completion of the group $B/\Delta$ is then $\{0\}$; applying Lemma 2 of no. 12 to the exact sequences $0 \to A \xrightarrow{u} B, A \xrightarrow{u} B \to B/\Delta$, we see that $\hat{u}$ is an isomorphism of $\hat{A}$ onto $\hat{B}$.