Classical Zariski Main from relative integral-closure neighbourhoods

Construct $N$ and $j$ by [F1]; their composite is $f$, since evaluation on the image of a base function is its pullback by $f$. Fix $x\in X$ and $y=f(x)$. By [F4], after a chosen elementary change $(T,t)\to (Y,y)$ the source decomposes as $X_T=V\bigsqcup W$, where $V\to T$ is finite and $V_t$ consists of the selected lift of $x$. Work on an affine neighbourhood of $t$ if necessary. Flat base change of sections in [F1] and integral-closure compatibility [F3] identify the relative normalization of $T$ in $X_T$ with $N_T=N{\times }_{Y}T$. The section algebra of the disjoint union is a product, and relative integral closure in a product is the product of the closures: one inclusion follows by projection of monic equations, and for the other multiply finitely many monic equations annihilating the two components. Since the algebra of $V$ is finite, it is already integral over the base. Therefore $N_T=V\bigsqcup N_W$, with $j_T{\mathrm{\mid }}_V$ the identity and $j_T(W)\subseteq N_W$. In particular $j_T^{-1}(V)=V$ and this restriction is an isomorphism.

The projection $N_T\to N$ is an elementary change by [F2], hence open. Let $H$ be the image of the open piece $V\subseteq N_T$; it is an open neighbourhood of $j(x)$. The map $p:V\to H$ is surjective and locally flat, and the square with vertical maps $j$ and $j_T$ identifies $X_H{\times }_{H}V$ with $V$, by step 1.1. We prove below that this forces $X_H\to H$ to be an isomorphism, using only affine algebra and the sheaf of regular functions. This is the nonaffine descent step; no chartwise factorization is assumed to glue by itself.

Fix a classical point $h\in H$ and a classical point $v\in V$ above it. Put $A={\mathcal{O}}_{H,h}$ and $B={\mathcal{O}}_{V,v}$. The local map $A\to B$ is faithfully flat by [F2]. Restricting the square of step 2.1 to these local affine models gives $X_A{\times }_A\mathrm{Spec}B\cong \mathrm{Spec}B$. Here $X_A$ is obtained by localizing the finite affine source cover at the base denominators; its charts and overlaps are ordinary localized rings. Its closed fibre has exactly one point: the local elementary-change fibre $B\mathrm{/}{\mathfrak{m}}_{A}B$ is the selected zero-dimensional regular fibre factor, hence equals $k$ by [F2]. Reducing the displayed isomorphism by ${\mathfrak{m}}_A$ therefore identifies that fibre with $\mathrm{Spec}k$. Choose an affine open $Q\subseteq X_A$ containing this point. The pullback $Q_B$ is open in $\mathrm{Spec}B$ and contains its closed point. Every open neighbourhood of the closed point of a local spectrum is the entire spectrum, so $Q_B=\mathrm{Spec}B$.

Write $Q=\mathrm{Spec}D$. The isomorphism of step 3.1 says $D{\otimes }_{A}B=B$ by the natural algebra map. Flatness and [F5] annihilate the kernel and cokernel of $A\to D$, so $A\to D$ is an isomorphism. The complement $X_A\setminus Q$ has empty pullback to $B$. If it contained a point with prime $\mathfrak{q}$ in an affine source chart, faithful flatness after tensoring with the residue field at $\mathfrak{q}$ would give a point above it; equivalently use the surjectivity on spectra in [F5] for that chart base change. Thus the complement is empty and $X_A\cong \mathrm{Spec}A$. In particular $j$ has exactly one point over $h$ and an isomorphism of its local rings there. This proves these assertions for every classical point of $H$.

The map $j$ is open onto its image. For an open $O\subseteq X$ and a point $x\in O$, use the construction of step 1.1. The open subset $O_T\cap V$ of $N_T$ projects to an open subset of $N$ containing $j(x)$, by [F2]. Every point of this projection is the image under $j$ of a point of $O$, because $j_T$ is the identity on $V$ and the square is Cartesian. Conversely every $j(x)$ with $x\in O$ belongs to one of these projections. Their union is therefore $j(O)$, which is open. The descent argument of steps 3.1–4.1 shows that over every $H$ of step 2.1 the map is bijective and induces isomorphisms on local rings. These $H$ cover $j(X)$, so $j$ is a homeomorphism onto the open subset $j(X)$ with an isomorphism of sheaves: a sheaf morphism whose stalk maps are isomorphisms is an isomorphism, as local inverses agree on overlaps. Thus $j$ is an open immersion. Since $\nu$ is finite by [F1], the required factorization follows.

If $X$ is empty, the section algebra and its integral closure are zero on every chart, so $N$ is empty and the factorization is immediate. Reducible cases were retained in [F1]; after elementary change they remain reduced by [F2]. The finite-type and finite-fibre hypotheses were used in [F1] and [F4], and separatedness was used to make the finite neighbourhood closed in [F4]. AC is inherited from the algebraic localization, flat prime-lifting and standard-smooth suppliers. Perfectness is not needed for this factorization; it remains necessary in the separate regular-to-smooth curve consequence. No later scheme theorem is used as a premise.