Normalization is unchanged under finite birational maps of reduced curves

By the stated birationality convention, each reduced component $X_i$ corresponds to exactly one reduced component $Y_j$, with the same generic field. The restriction $f_i:X_i\to Y_j$ exists: the ideal of $Y_j$ pulls back to zero on the generic point of the reduced integral scheme $X_i$, hence to zero everywhere on $X_i$. It is finite, since on affine charts its coordinate algebra is a quotient of the finite $A$-algebra for $f$, and the $A$-action factors through the quotient defining $Y_j$. Thus $f_i$ is finite and birational in the integral sense of [F7].

The affine normalization construction of [F1] separates the reduced components, so $\tilde{X}={\coprod }_i{\tilde{X}}_i$ and $\tilde{Y}={\coprod }_j{\tilde{Y}}_j$. Indeed for a reduced Noetherian affine curve with minimal primes ${\mathfrak{p}}_i$, its total ring of fractions is ${\prod }_i\mathrm{Frac}(A\mathrm{/}{\mathfrak{p}}_i)$, as established in the construction of [F1]. Its integral closure is ${\prod }_i\overline{A\mathrm{/}{\mathfrak{p}}_i}$: projection of a monic equation proves one inclusion; conversely, lift a monic equation for each coordinate to $A[T]$ and multiply the finitely many lifted polynomials, obtaining a monic equation annihilating the tuple. These identifications commute with restrictions and give the claimed decompositions. It therefore suffices to compare the normalizations for each $f_i$.

Fix $i,j$ and an affine open $U=\mathrm{Spec}A\subseteq Y_j$ with $f_i^{-1}(U)=\mathrm{Spec}B$; then $A$ and $B$ are domains of dimension one, finite type over $k$, the map $A\to B$ is injective, module-finite by [F2] and integral by [F3], and the birationality of $f_i$ gives $\mathrm{Frac}(A)=\mathrm{Frac}(B)$ as subfields of the common function field $K(Y_j)=K(X_i)$. Indeed [F4] applied to $f_i$ supplies a nonempty affine open of $Y_j$ on which the localised map is an isomorphism, and localising a domain at a nonzero element does not change its fraction field. By [F1] the normalization ${\tilde{Y}}_j$ over $U$ is the spectrum of the integral closure $\bar{A}$ of $A$ in $\mathrm{Frac}(A)$ and ${\tilde{X}}_i$ over $f_i^{-1}(U)$ is the spectrum of the integral closure $\bar{B}$ of $B$ in $\mathrm{Frac}(B)$ ([F5]).

In the situation of step 3.1 one has $\bar{A}=\bar{B}$ as subrings of the common field $\mathrm{Frac}(A)=\mathrm{Frac}(B)$: since $A\subseteq B\subseteq \mathrm{Frac}(B)$ and $B$ is integral over $A$, [F6] says that an element is integral over $A$ exactly when it is integral over $B$. Hence the affine normalizations agree canonically over $U$, and the identification is the identity on the common function field.

The identifications of step 4.1 are canonical on affine charts (both sides are the same integral closure inside the same function field), so they agree on overlaps and glue to an isomorphism ${\tilde{X}}_i\to {\tilde{Y}}_j$ over $Y_j$; assembling over the components by step 2.1 gives the isomorphism $\tilde{X}\to \tilde{Y}$ over $Y$, and the uniqueness clause of [F1] makes it the canonical identification of the two normalizations with the same finite birational model of $Y$.