The finite normalization of a curve factors through the blowup of a closed point

The blowup $\beta$ is finite and is an isomorphism over $Y\setminus \{p\}$ by [F2]; in particular $\beta$ is affine and ${\beta }_{\ast }{\mathcal{O}}_{Y_1}$ is a coherent ${\mathcal{O}}_Y$-module. The scheme $Y^{\nu }$ is normal of dimension one, so each of its local rings is a discrete valuation ring or a field by [F1].

The normalization is integral with the same function field as $Y$. At a point $q$ above $p$, the map ${\mathcal{O}}_{Y,p}\to {\mathcal{O}}_{Y^{\nu },q}$ embeds both rings into that function field. The maximal ideal ${\mathfrak{m}}_p$ contains a nonzero element, whose image remains nonzero; its extended ideal is proper because the map of local rings is local. The point $q$ is closed, since the fibre of the finite morphism $\nu$ is zero-dimensional, and its normal local ring is therefore a DVR by step 1.1. Every nonzero ideal in a DVR is generated by a nonzerodivisor, so ${\mathfrak{m}}_p{\mathcal{O}}_{Y^{\nu },q}$ is principal and invertible. At points not over $p$ the pullback center ideal is the unit ideal. This coherent ideal is thus locally invertible everywhere: local stalk generators extend to neighbourhoods, and the equality with the principal ideal holds after shrinking because its cokernel is coherent. Its inverse-image subscheme is an effective Cartier divisor.

By the universal property [F3] applied to the center $\{p\}\subseteq Y$ and the morphism $\nu :Y^{\nu }\to Y$ (whose inverse image of $p$ is effective Cartier by step 2.1), there is a unique $Y$-morphism ${\nu }_1:Y^{\nu }\to Y_1$ with $\beta \circ {\nu }_1=\nu$. This morphism is dominant: $\nu$ is surjective and $\beta$ is an isomorphism over $Y\setminus \{p\}$, so ${\nu }_1(Y^{\nu })\supseteq {\beta }^{-1}(Y\setminus \{p\})$, a nonempty open subset of the irreducible scheme $Y_1$ by [F4] and therefore dense.

The morphism ${\nu }_1$ is finite: over an affine open $U=\mathrm{Spec}R\subseteq Y$, write $Y_1{\mathrm{\mid }}_U=\mathrm{Spec}B$ and $Y^{\nu }{\mathrm{\mid }}_U=\mathrm{Spec}C$ with $R\to B$ and $R\to C$ module-finite by [F2]; the factorization gives a ring map $B\to C$, and since $C$ is a finitely generated $R$-module with $R\subseteq B$ acting through $B\to C$, the ring $C$ is a finitely generated $B$-module; hence ${\nu }_1$ is affine with module-finite coordinate algebras, i.e. finite (Finite morphisms of schemes).

Finally, if ${\mathcal{O}}_{Y,p}$ is regular, then $\beta$ is an isomorphism by [F2], and under the identification $Y_1=Y$ the unique factorization ${\nu }_1$ of $\nu$ through the identity is $\nu$ itself, by uniqueness in step 3.1; the two displayed clauses about the regular case follow.

The morphism ${\nu }_1$ is birational: $\nu$ is an isomorphism over a dense open $V\subseteq Y$ (birationality), and $\beta$ is an isomorphism over $Y\setminus \{p\}$ by [F2]; hence ${\nu }_1$ is an isomorphism over the dense open ${\beta }^{-1}(V\setminus \{p\})$ of $Y_1$ (here $V\setminus \{p\}$ is nonempty open in the one-dimensional irreducible scheme $Y$, hence dense). Since $Y_1$ is integral by [F4] and $Y^{\nu }$ is normal, ${\nu }_1$ is a finite birational morphism from a normal scheme onto the integral scheme $Y_1$. The affine coordinate ring $C$ of the source is the integral closure of the coordinate ring $B$ of $Y_1$ in their common function field: $C$ is integral over $B$ by finiteness, while every element integral over $B$ is also integral over $C$ and therefore belongs to the integrally closed ring $C$. Thus ${\nu }_1$ is the normalization of $Y_1$.

On an affine chart $V=\mathrm{Spec}B$ of the integral scheme $Y_1$, the inverse image under the finite birational normalization map is $\mathrm{Spec}C$. The map $B\to C$ is injective: localizing it at the generic point is the identified inclusion of the common function field, so any element in its kernel is zero in $\mathrm{Frac}(B)$ and hence zero in the domain $B$. Thus ${\mathcal{O}}_{Y_1}\hookrightarrow {\nu }_{1,\ast }{\mathcal{O}}_{Y^{\nu }}$. Applying the left exact pushforward along $\beta$ gives ${\beta }_{\ast }{\mathcal{O}}_{Y_1}\hookrightarrow {\nu }_{\ast }{\mathcal{O}}_{Y^{\nu }}$. Coherence follows from finiteness over the Noetherian base, establishing the claimed coherent subalgebra.

Steps 1.1-6.1 and 4.2 prove: $\beta$ is finite; $\nu$ factors uniquely as $\beta \circ {\nu }_1$ with ${\nu }_1:Y^{\nu }\to Y_1$; $Y^{\nu }$ is a normalization of $Y_1$; and ${\beta }_{\ast }{\mathcal{O}}_{Y_1}$ is a coherent ${\mathcal{O}}_Y$-subalgebra of ${\nu }_{\ast }{\mathcal{O}}_{Y^{\nu }}$, with the regular case giving $\beta ={\mathrm{id}}_Y$ and ${\nu }_1=\nu$ under $Y_1=Y$.