Theorem 4 (Serre)

Let $A$ be a noetherian ring satisfying the following conditions: (i) for every minimal prime ideal $\mathfrak{p}$ of $A$, the ring $A_{\mathfrak{p}}$ is reduced; (ii) for every prime ideal $\mathfrak{p}$ of $A$ of height 1, the ring $A_{\mathfrak{p}}$ is integrally closed; (iii) for every prime ideal $\mathfrak{p}$ of $A$ of height $\geqslant 2$, one has $\operatorname{prof}(A_{\mathfrak{p}}) \geqslant 2$.

Then $A$ is normal.

It is a question of proving that the ring $A_{\mathfrak{p}}$ is integrally closed for every $\mathfrak{p} \in \operatorname{Spec}(A)$, which we shall do by induction on $\operatorname{ht}(\mathfrak{p})$. For $\operatorname{ht}(\mathfrak{p}) \leqslant 1$ this follows from assumptions (i) and (ii). Suppose therefore that one has $\operatorname{ht}(\mathfrak{p}) \geqslant 2$ and that $A_q$ is integrally closed for every prime ideal $q$ of $A$ of height $< \operatorname{ht}(\mathfrak{p})$. By assumption (iii), one has $\operatorname{prof}(A_{\mathfrak{p}}) \geqslant 2$. By the induction hypothesis and cor. 3 of th. 3 of No. 9, applied to the ring $A_{\mathfrak{p}}$ and to the closed subset $\{\mathfrak{p}A_{\mathfrak{p}}\}$ of $\operatorname{Spec}(A_{\mathfrak{p}})$, the ring $A_{\mathfrak{p}}$ is an integral domain. Let $K$ be its field of fractions and let $B$ be a subring of $K$, containing $A_{\mathfrak{p}}$ and finite over $A_{\mathfrak{p}}$. It is a question of proving that $B$ is equal to $A_{\mathfrak{p}}$. Let us denote by $i$ the canonical injection of $A_{\mathfrak{p}}$ into $B$. Since $B$ is contained in $K$, it is a torsion-free $A_{\mathfrak{p}}$-module, hence of depth $\geqslant 1$. Moreover, for every prime ideal $q$ of $A_{\mathfrak{p}}$ distinct from $\mathfrak{p}A_{\mathfrak{p}}$, the homomorphism $i_q : (A_{\mathfrak{p}})_q \to B_q$ is bijective since $A_q$ is integrally closed. By lemma 4 of No. 9, applied to the closed subset $F = \{\mathfrak{p}A_{\mathfrak{p}}\}$ of $\operatorname{Spec}(A_{\mathfrak{p}})$, the homomorphism $i$ is bijective, which completes the proof of the theorem.