Corollary 2

*Let $\rho : A \to B$ be a homomorphism of noetherian rings making $B$ into a flat $A$-module.

a) If $B$ is normal and faithfully flat over $A$, $A$ is normal.

b) Suppose that $A$ is normal and that the ring $\kappa(p) \otimes_A B$ is normal when $p$ is a minimal prime ideal of $A$, and reduced when $p$ is a prime ideal of height 1. Then the ring $B$ is normal.

a) Suppose $B$ normal and faithfully flat over $A$. Then $\rho$ is injective (I, § 3, No. 5, Prop. 9) and $B$ is reduced, hence $A$ is reduced. Let $S$ be the set of non-zero-divisors of $A$. Since $B$ is flat over $A$, $\rho(S)$ consists of non-zero-divisors in $B$, so that $B$ is integrally closed in $\rho(S)^{-1}B$. Let $x$ be an element of $S^{-1}A$ integral over $A$; then the element $x \otimes 1_B$ of the ring $S^{-1}A \otimes_A B$ (which is identified with $\rho(S)^{-1}B$) is integral over $B$, hence belong to $B$. Since $B$ is faithfully flat over $A$, one has $x \in A$ (I, § 3, No. 5, Prop. 10, (ii)), and $A$ is normal.

b) Under the hypotheses of b), it is enough, by Remark 1, to prove that for every prime ideal $q$ of $B$, the local ring $B_q$ is normal or of depth $\geqslant 2$. Let $p$ denote the prime ideal $\rho^{-1}(q)$ of $A$; the local homomorphism $A_p \to B_q$ deduced from $\rho$ makes $B_q$ into a faithfully flat $A_p$-module (I, § 3, No. 5, Prop. 9). Let us distinguish four cases :

1) $\mathrm{ht}(p) = 0$. Then $A_p$ is a field, equal to $\kappa(p)$; the ring $A_p \otimes_A B$ is normal by assumption. The same is true of $B_q$, which is a ring of fractions thereof.