Lemma 1

Let $A$ be a noetherian ring, the union of an increasing filtered family $(A_\alpha)_{\alpha \in I}$ of noetherian subrings.

a) If the rings $A_\alpha$ are regular and if $A$ is a flat $A_\alpha$-module for every $\alpha \in I$, the ring $A$ is regular.

b) If the rings $A_\alpha$ are normal ($§ 1$, No. 8), $A$ is normal.

a) Let $m$ be a maximal ideal of $A$; for each $\alpha \in I$, let us denote by $m_\alpha$ the ideal $m \cap A_\alpha$ of $A_\alpha$. Since $A$ is noetherian, $m$ is of finite type and there exists an element $\alpha$ of $I$ such that $m = A m_\alpha$, so that the $A$-module $(A_\alpha / m_\alpha) \otimes_{A_\alpha} A$ is isomorphic to $A / m$. Since $A$ is flat over $A_\alpha$, we have $\mathrm{dp}_A(A/m) \leq \mathrm{dp}_{A_\alpha}(A_\alpha / m_\alpha)$ (A, X, p. 141, Lemma 2). Since the rings $A_\alpha$ are regular, it follows that $A$ is regular ($§ 4$, No. 2, Prop. 4).

b) Since the $A_\alpha$ are reduced, $A$ is reduced. Let $a$ and $b$ be elements of $A$ such that $b$ is not a zero divisor and that the element $a/b$ of the total ring of fractions of $A$ is integral over $A$. There exists a monic polynomial $P \in A[X]$ such that $P(a/b) = 0$. Let $\alpha$ be an element of $I$ such that the ring $A_\alpha$ contains $a$, $b$ and the coefficients of $P$. Since $A_\alpha$ is normal, there exists $c \in A_\alpha$ such that $a = bc$. Thus $a/b = c$ belongs to $A$ and $A$ is normal.