Proposition 1

a) The ring A is local, with maximal ideal $m = \lim \overrightarrow{m_\alpha}$. For every $\alpha \in I$, the homomorphism $\varphi_\alpha$ is local and flat, and one has $\varphi_\alpha(m_\alpha)$ $A = m$.

b) If $A_\alpha$ is noetherian for every $\alpha \in A$, then A is noetherian.

a) Put $m = \lim \overrightarrow{m_\alpha}$; this is an ideal of A. The quotient ring $A/m$ is the inductive limit of the fields $A_\alpha/m_\alpha$, hence is a field (A, I, p. 116, Prop. 3). On the other hand, every element of $A - m$ is invertible in A: in fact, let $x \in A - m$; there exist $\alpha \in I$ and $\xi \in A_\alpha$ such that $x = \varphi_\alpha(\xi)$; one has $\xi \notin m_\alpha$, hence $\xi$ is invertible in $A_\alpha$ and x is invertible in A. Consequently, A is a local ring, with maximal ideal m. Let $\alpha \in I$. From the relations $\varphi_{\beta\alpha}(m_\alpha)$ $A_\beta = m_\beta$ for $\beta \geq \alpha$, one deduces, by passing to the inductive limit, $\varphi_\alpha(m_\alpha)$ $A = m$; finally, the homomorphism $\varphi_\alpha$ is flat by I, § 2, No. 7, Prop. 9.

(b) Let $\hat{A}$ be the separated completion of A for the $m$-adic topology and $\pi$ the canonical mapping of A into $\hat{A}$. Suppose that the rings $A_\alpha$ are noetherian. Fix $\alpha \in I$ and prove that the ring $\hat{A}$ is noetherian and flat over $A_\alpha$. By hypothesis, $m_\alpha$ is a finitely generated ideal of $A_\alpha$, hence $m = \varphi_\alpha(m_\alpha)$. A is a finitely generated ideal of A. It follows that the maximal ideal $\hat{m}$ of $\hat{A}$ is equal to $mA$ (III, § 2, No. 12, Cor. 2 to Prop. 16 and No. 13, Prop. 19), hence is finitely generated. Consequently, the ring $\hat{A}$ is noetherian (loc. cit., No. 10, Cor. 5 to Th. 2). On the other hand, for every $n \in \mathbf{N}$, the quotient $\hat{A}/\hat{m}^n$ is isomorphic to $A/m^n$ (loc. cit., No. 12, Cor. 2 to Prop. 16 and formula (21)), which means that $\hat{A}/\pi \circ \varphi_\alpha(m_\alpha^n)$ $\hat{A}$ is isomorphic to $A \otimes_A (A_\alpha/m_\alpha^n)$; since A is a flat $A_\alpha$-module, the $(A_\alpha/m_\alpha^n)$-module $\hat{A}/\pi \circ \varphi_\alpha(m_\alpha^n)$ $\hat{A}$ is flat for every $n \in \mathbf{N}$. By III, § 5, No. 4, Prop. 2, the $A_\alpha$-module $\hat{A}$ is ideally separated for $m_\alpha$; by loc. cit., No. 2, Th. 1, the $A_\alpha$-module $\hat{A}$ is therefore flat. It follows by passing to the inductive limit that $\hat{A}$ is (faithfully) flat over A (I, § 2, No. 7, Prop. 9), hence that A is noetherian (I, § 3, No. 5, corollary to Prop. 8).