Normalization of an affine integral curve

By [F1], $B$ is a subring of $K$ containing $A$ and a finite $A$-module. It is therefore a domain, and its fraction field is $K$: it lies inside $K$ and contains $A$. It is also a finite-type $k$-algebra, since a finite set of $A$-module generators generates it as an $A$-algebra and $A$ is finite type over $k$. The inclusion $A\subseteq B$ is integral by the definition of $B$. Its spectrum is affine by construction, and the one-affine-cover criterion in [F2] makes $\nu$ finite.

The ring $B$ is integrally closed in $K$: if $x\in K$ is integral over $B$, transitivity in [F1] makes $x$ integral over $A$, so $x\in B$. For any prime $\mathfrak{q}$ of $B$, the localization $B_{\mathfrak{q}}$ is integrally closed in $K$. Indeed, if $x\in K$ satisfies a monic equation over $B_{\mathfrak{q}}$, choose $s\in B\setminus \mathfrak{q}$ clearing all coefficient denominators. Multiplying the equation of degree $n$ by $s^n$ shows that $sx$ satisfies a monic equation over $B$; hence $sx\in B$ and $x\in B_{\mathfrak{q}}$. Thus $\mathrm{Spec}B$ is normal.

By [F2], $\dim B=\dim A=1$. The scheme $\mathrm{Spec}B$ is integral, separated and finite type over $k$ by step 1.1, and normal by step 2.1. The perfect-field theorem [F3] therefore makes it smooth over $k$.

Suppose $k$ is algebraically closed. For either $R=A$ or $R=B$, write $R$ as a quotient of a finite polynomial algebra over $k$. A maximal ideal of $R$ pulls back to a maximal ideal of that polynomial algebra, whose quotient is $k$ by [F5]. Consequently every closed point of $\mathrm{Spec}R$ has residue field $k$.

Let $\mathfrak{m}$ be a maximal ideal of $A$. By [F4], there is a prime of $B$ over $\mathfrak{m}$, and every such prime is maximal. The quotient $C=B\mathrm{/}\mathfrak{m}B$ is a finite-dimensional vector space over $A\mathrm{/}\mathfrak{m}=k$, because $B$ is a finite $A$-module. Its prime ideals are exactly the primes of $B$ over $\mathfrak{m}$, and they are maximal. If ${\mathfrak{n}}_1,\dots ,{\mathfrak{n}}_r$ are distinct such ideals in $C$, they are pairwise comaximal; [F5] gives a surjection $C\to {\prod }_{i}C\mathrm{/}{\mathfrak{n}}_i$. Each factor is a nonzero $k$-vector space, so $r\le {\dim }_{k}C$. Thus there can be only finitely many points in the fiber, and it is nonempty. By step 4.1 both residue fields at each such point are $k$, and their map is a $k$-algebra map, hence the identity under these identifications. This proves all assertions.