Corollary 1

Let $A$ be a noetherian ring, $F$ a closed subset of $\mathrm{Spec}(A)$, $U$ the complementary open set. Suppose that $\mathrm{prof}_F(A)$ is $\geqslant 2$ (resp. $\geqslant 1$) and that, for every $p \in U$, the ring $A_p$ is integrally closed (resp. reduced). Then $A$ is normal (resp. reduced).

For every $p \in F$, one has $\mathrm{prof}(A_p) \geqslant \mathrm{prof}_F(A)$ (No. 5, Prop. 8); it is therefore enough to apply the preceding remark.