Normalization of a reduced curve is finite

Since $C$ is quasi-compact and its affine coordinate rings are Noetherian, it is Noetherian. Its finitely many reduced irreducible components $C_i$ are integral finite-type curves of dimension one. Choose a finite affine cover $U_{ij}=\mathrm{Spec}{A}_{ij}$ of each component and identify all their fraction fields with its generic field $K_i$. Let $B_{ij}$ be the integral closure of $A_{ij}$ in $K_i$. These are finite Noetherian normal domains.

The schemes $\mathrm{Spec}{B}_{ij}$ glue over $C_i$, even if $C_i$ is nonseparated. Here are the overlap identifications explicitly. For a point in $U_{ij}\cap U_{il}$ choose distinguished neighborhoods $D(f)\subset U_{ij}$ and $D(g)\subset U_{il}$ contained in that intersection. On $D(f)$ write $g=c\mathrm{/}{f}^r$, and on $D(g)$ write $f=d\mathrm{/}{g}^s$. Their common intersection is the distinguished open $D(fc)$ in $U_{ij}$ and $D(gd)$ in $U_{il}$; both coordinate rings are the same subring of $K_i$, since they are the sections of the same open subscheme. Their integral closures are therefore the same subring of $K_i$, namely $(B_{ij}{)}_{fc}=(B_{il}{)}_{gd}$. These common distinguished opens cover the overlap. Their identities glue, and the triple-overlap identities satisfy the cocycle condition because all are identities inside $K_i$. Thus scheme and morphism gluing produce an integral normal scheme $C_i^{\mathrm{n}\mathrm{u}}$ and a morphism ${\nu }_i:C_i^{\mathrm{n}\mathrm{u}}\to C_i$, with inverse image of $U_{ij}$ equal to $\mathrm{Spec}{B}_{ij}$. It is finite by the affine-cover criterion. Its generic field is $K_i$, and its dimension is one by integral dimension preservation on the affine cover.

Set $C^{\mathrm{n}\mathrm{u}}={⨆}_i{C}_i^{\mathrm{n}\mathrm{u}}$ and compose each ${\nu }_i$ with the reduced closed immersion $C_i\hookrightarrow C$ to obtain $\nu$. This is finite: closed immersions are finite on affine charts by the quotient-ring description; composition is module-finite, and a finite disjoint union is module-finite. Empty affine opens have empty inverse image, with coordinate algebra zero; the following computation concerns nonempty affine opens. More explicitly, on any affine $U=\mathrm{Spec}A\subset C$, let ${\mathfrak{p}}_1,\dots ,{\mathfrak{p}}_m$ be its minimal primes and $A_i=A\mathrm{/}{\mathfrak{p}}_i$. The corresponding component inverse images are affine with finite coordinate domains $D_i$ lying in $\mathrm{Frac}(A_i)$, normal and birational over $A_i$. Consequently $D_i$ is its integral closure $B_i$: every element of $D_i$ is integral over $A_i$, and every element of the fraction field integral over $A_i$ is also integral over $D_i$, hence belongs to $D_i$. Therefore ${\nu }^{-1}(U)=\mathrm{Spec}B$ where $B={\prod }_i{B}_i$. Each factor has finitely many $A_i$-module generators; placing these in their separate coordinates gives finitely many $A$-module generators of $B$. This uses no Chinese-remainder decomposition of $A$.

We identify $Q(A)$ correctly. Distinct minimal primes are incomparable. For each $i$ choose $a_{ij}\in {\mathfrak{p}}_j\setminus {\mathfrak{p}}_i$ for $j\mathrm{\ne }i$ and put $g_i={\prod }_{j\mathrm{\ne }i}{a}_{ij}$; for one minimal prime put $g_1=1$. Its image is nonzero in $A_i$ and zero in every other $A_j$. An element $s\in A$ is a nonzerodivisor exactly when its image is nonzero in every $A_i$: the forward implication follows since otherwise $s{g}_i=0$ with $g_i\mathrm{\ne }0$, and the reverse follows from the injection $A\hookrightarrow {\prod }_i{A}_i$. Hence localization gives an injection $Q(A)\hookrightarrow {\prod }_i\mathrm{Frac}(A_i)$. For any tuple $u_i\mathrm{/}{v}_i$ in this product, choose lifts $a_i,b_i\in A$ of $u_i,v_i$, and set $a={\sum }_i{g}_i{a}_i$ and $s={\sum }_i{g}_i{b}_i$. The image of $s$ in $A_i$ is the nonzero product $g_i{v}_i$, so $s$ is a nonzerodivisor, and $a\mathrm{/}s$ has the prescribed tuple of images. This proves surjectivity.

Each $C_i^{\mathrm{n}\mathrm{u}}$ is normal Noetherian of dimension one. All its local rings are therefore fields or DVRs, hence regular. Their disjoint union is regular of dimension one. The generic-field identifications give birationality on each component, meaning the restriction from the corresponding normalized component, not a claim that scheme-theoretic base change over $C_i$ removes the other branches.

The affine description gives a finite module $B$ on each affine target chart. Thus ${\nu }_{\ast }{\mathcal{O}}_{C^{\mathrm{n}\mathrm{u}}}$ is quasi-coherent of finite type, hence coherent on the locally Noetherian scheme $C$.

Under that identification $B={\prod }_i{B}_i$ is the integral closure of $A$ in $Q(A)$. Since $B$ is module-finite over $A$, every $b\in B$ satisfies a monic equation over $A$: multiplication by $b$ on a finite generating family and the determinant trick give a monic polynomial annihilating $B$, hence annihilating $1$. Conversely a tuple integral over $A$ has its $i$th coordinate integral over $A_i$, so that coordinate belongs to $B_i$. This proves precisely the affine description in part (3).

At a regular point $x$ of $C$, its local ring is a field (in dimension zero its maximal ideal has zero cotangent space and hence is zero by Nakayama) or a DVR, hence an integrally closed domain. Thus only one component passes through $x$. Remove the other finitely many closed components to obtain a neighborhood with integral coordinate rings. For an affine neighborhood $\mathrm{Spec}A$ therein, localization of its integral closure at the prime of $x$ equals $A_x$. Indeed any element integral over $A_x$ has an equation with finitely many denominators outside that prime; clearing these denominators after multiplying the element by their product shows it belongs to a localization of the integral closure of $A$. The reverse inclusion is immediate. As $B\mathrm{/}A$ is a finite module, its zero stalk at $x$ implies it vanishes on a distinguished neighborhood of $x$ (annihilate each of finitely many generators with an element outside the prime). On that neighborhood $A=B$, so $\nu$ is an isomorphism. These neighborhoods cover the regular locus.

Any other morphism with the stated properties has, by part (3), the same integral-closure algebra $B\subset Q(A)$ on each affine target chart. These canonical identifications commute with restriction: they are the same identifications inside the component function fields used in step 2.1. They therefore glue to a $C$-isomorphism. It is unique: an $A$-algebra automorphism of $B$ localizes to an automorphism of $Q(A)$ fixing $A$, hence fixing every fraction $a\mathrm{/}s$. Here localization of $B$ at the nonzerodivisors of $A$ equals $Q(A)$, since $A\subset B\subset Q(A)$. Since $B$ embeds in $Q(A)$, that automorphism is already the identity on $B$. This proves part (4).