Proposition 11

Let $\mathbf{A}$ be a Noetherian local ring, $m$ its maximal ideal, $\hat{\mathbf{A}}$ its $m$-adic completion and $B$ a ring such that $\mathbf{A} \subset B \subset \hat{\mathbf{A}}$. Suppose that $B$ is a Noetherian local ring whose maximal ideal $n$ satisfies the relation $n = mB$. Then

for all $k \geq 1$, the $n$-adic topology on $B$ is induced by the $m$-adic topology on $\hat{\mathbf{A}}$, $B$ is a faithfully flat $\mathbf{A}$-module and there is an isomorphism of $\hat{\mathbf{A}}$ onto the $n$-adic completion $\hat{B}$ of $B$, which extends the canonical injection $\mathbf{A} \to B$.

It is sufficient to verify the relation $n^k = \hat{m}^k \cap B$, for, as $B$ is dense in $\hat{\mathbf{A}}$ and the $n$-adic topology is induced by the $m$-adic topology, the last assertion will follow from General Topology, Chapter II, § 3, no. 9, Corollary 1 to Proposition 18 and the last but one from Proposition 10. The injection $j_A : \mathbf{A} \to \hat{\mathbf{A}}$ (resp. $j : B \to \hat{\mathbf{A}}$) defines by taking quotients an injective homomorphism

(resp. $i_B : B/(\hat{m} \cap B) \to \hat{\mathbf{A}}/\hat{m}$). We know that $m \cap \mathbf{A} = m$ and that $i_A$ is bijective, hence $i_B$ is bijective, which shows that $B/(\hat{m} \cap B)$ is a field, hence that $\hat{m} \cap B$ is a maximal ideal of $B$ and therefore $m \cap B = n$. As $\mathbf{A} = \mathbf{A} + m$, $B = A + n = A + mB$; by induction on $k$ we deduce that

for all $k > 1$. As $n^k \subset \hat{m}^k \cap B$, it is sufficient to show that $\hat{m}^k \cap B \subset n^k$; if $b \in \hat{m}^k \cap B$, we may write $b = a + z$ where $a \in \mathbf{A}$, $z \in n^k$; whence

and $b \in n^k$.

\* An important case where this applies is the following: $B$ is the ring of integral series in $n$ variables over a complete valued field, which converge in the neighbourhood of 0, $\mathbf{A}$ is the local ring

where $\mathfrak{p}$ is the maximal ideal consisting of the polynomials with no constant term and $\hat{\mathbf{A}}$ is the ring of formal power series $K[[X_1, \ldots, X_n]]$. \*