Surface finite type normalization finite

At every maximal ideal $\mathfrak{m}$ of $B$ the formal-fibre lemma makes the completion of $B_{\mathfrak{m}}$ reduced: flatness embeds the completion into its generic fibre, which is geometrically regular and hence reduced.

For the reduced complete equicharacteristic local ring $T$, let ${\mathfrak{p}}_i$ be its finitely many minimal primes. Each $T\mathrm{/}{\mathfrak{p}}_i$ is a complete local domain with finite normalization. The injection $T\to {\prod }_{i}T\mathrm{/}{\mathfrak{p}}_i$ identifies the total quotient ring with ${\prod }_i\mathrm{Frac}(T\mathrm{/}{\mathfrak{p}}_i)$: prime avoidance isolates the finitely many components after inverting nonzerodivisors. The product of their normalizations is finite over $T$ and is integrally closed in this total quotient ring. It is integral over $T$, and any element integral over $T$ is integral over this product and hence lies in it. Thus it is the finite normalization of $T$. The ring $T$ itself need not be a product of domains.

If $C$ is the normalization of $B_{\mathfrak{m}}$ and $T$ the completion, then $C{\otimes }_{B_{\mathfrak{m}}}T$ embeds into the total quotient ring of $T$, is integral over $T$ and hence a $T$-submodule of the finite normalization of step 2.1, so it is a finite $T$-module; faithful flatness of completion descends finitely many generators and makes $C$ finite over $B_{\mathfrak{m}}$.

Choose $0\mathrm{\ne }f\in B$ with $B_f$ regular by the open-regular-locus supplier. Any finite birational $B\subset B^{\mathrm{\prime }}\subset \mathrm{Frac}B$ has $B_f^{\mathrm{\prime }}=B_f$. Its nonnormal locus is closed: outside $V(f)$ it is regular; on $V(f)$ the height-one primes are among the finitely many minimal primes of $(f)$, and the $S_2$ obstructions are among the associated primes of $B^{\mathrm{\prime }}\mathrm{/}f{B}^{\mathrm{\prime }}$ of height at least two. At such a prime the multiplication-by-$f$ exact sequence and residue-field Ext give depth one: the domain has depth at least one, and its quotient has depth zero by [F10]. The same sequence gives depth at least two exactly when that quotient has no depth-zero stalk. Conversely, absence of these associated primes under a given prime gives $S_2$ for every localization of its local ring, and regularity at the height-one primes gives $R_1$. The Serre criterion therefore identifies the nonnormal locus with the union of the closures of those finitely many bad primes.

For each maximal ideal $\mathfrak{m}$ choose global integral elements whose localization generates the finite normalization of $B_{\mathfrak{m}}$. The finite algebra $B^{\mathrm{\prime }}$ they generate has normal localizations at every prime over $\mathfrak{m}$. Its closed nonnormal locus has closed image in $\mathrm{Spec}B$, since $B^{\mathrm{\prime }}$ is finite, and that image avoids $\mathfrak{m}$. On the complementary open, $B^{\mathrm{\prime }}$ equals the integral closure by normality and birationality. These opens cover all maximal ideals and hence all primes; quasi-compactness gives finitely many of them. The composite of their finite algebras is finite and equals the integral closure on every selected open, hence globally. Integral closure commutes with localization, and the affine constructions glue to finite normalization of schemes. AC and DC are inherited from the suppliers.