Relative normalization and its finite-stage reduction

On an affine open $U=\mathrm{Spec}R\subseteq S$, put $M_U=\mathrm{\Gamma }(f^{-1}U,{\mathcal{O}}_X)$ and let $C_U\subseteq M_U$ be the elements integral over the image of $R$. By [F1], on a principal open $D(r)\subseteq U$ one has $M_{D(r)}\cong (M_U{)}_r$. By [F2], the integral closure of $R_r$ in $(M_U{)}_r$ is $(C_U{)}_r$. Thus the $C_U$ agree on a basis of overlaps and glue to a quasi-coherent ${\mathcal{O}}_S$-algebra $\mathcal{C}\subseteq f_{\ast }{\mathcal{O}}_X$. This proves the first assertion about $\mathcal{C}$ without the later étale or finite-stage inputs.

The inclusion $\mathcal{C}\to f_{\ast }{\mathcal{O}}_X$ gives, by the relative spectrum adjunction in [F2], a canonical $S$-morphism $j:X\to N={\mathrm{Spec}}_S\mathcal{C}$. Every $C_U$ is integral over $R$ by definition, so $N\to S$ is affine and integral on each affine base chart. The construction is compatible with restriction to affine opens by step 1.1.

Let $x\in X$ over $s\in S$. By [F4], choose an elementary étale neighbourhood $(T,t)\to (S,s)$ on which the selected part of $X_T$ is an open-and-closed finite $T$-scheme $V$. By [F3], the integral closure algebra of $T$ in $(f_T{)}_{\ast }{\mathcal{O}}_{X_T}$ is the pullback of $\mathcal{C}$ on a neighbourhood of $t$. Shrink $T$ to that neighbourhood; the elementary étale property and the clopen finite decomposition persist, and the base-change identity now holds throughout $T$. The decomposition $X_T=V\bigsqcup W$ yields a product decomposition of this pushforward algebra. Since the algebra of the finite $T$-scheme $V$ is integral over ${\mathcal{O}}_T$, its integral closure in itself is itself; therefore the base-changed $N_T$ has a corresponding factor $V$, and $j_T$ is the identity on that factor. This is the local finite-component calculation; it uses no claim that the complementary $W$ is finite.

Use the actual étale neighbourhoods to prove openness. For each $x\in X$, step 3.1 gives a finite clopen factor $V\subseteq X_T$ identified by $j_T$ with a clopen factor of $N_T$. The projection $p:N_T\to N$ is étale and therefore open by [F8]. Its image $p(V)$ is an open neighbourhood of $j(x)$ contained in $j(X)$, because every point of $V$ comes from $X_T$. These images cover $j(X)$, making $j(X)$ open in $N$. For any open $O\subseteq X$, the sets $p(j_T(O_T\cap V))$ are open subsets of $j(O)$ and cover it as $x$ varies through $O$. Hence $j$ is an open map onto its open image. This argument uses the open sets $V$ in the actual étale cover $N_T$, rather than a local-ring isomorphism alone.

Fix $x\in X$ and $y=j(x)$. Set $A={\mathcal{O}}_{N,y}$, and for the chosen elementary étale chart $(T,t)$ set $A^{\mathrm{\prime }}={\mathcal{O}}_{N_T,(y,t)}$. The étale local map $A\to A^{\mathrm{\prime }}$ is flat and local, hence faithfully flat. By [F7], the chosen lift of $x$ is the unique point above $(x,t)$ and the chosen lift of $y$ is the unique point above $(y,t)$. The selected clopen factor $V\subseteq N_T$ contains $(y,t)$; hence its inverse image in $\mathrm{Spec}{A}^{\mathrm{\prime }}$ is all of that local scheme, since an open containing the closed point of a local scheme is the whole scheme. The complementary part of $X_T$ maps to a disjoint factor of $N_T$, and $j_T{\mathrm{\mid }}_V$ is the identity. Consequently the entire base change $X_A{\times }_{\mathrm{Spec}A}\mathrm{Spec}{A}^{\mathrm{\prime }}\to \mathrm{Spec}{A}^{\mathrm{\prime }}$ is an isomorphism, where $X_A=X{\times }_N\mathrm{Spec}A$.

Choose an affine open $W=\mathrm{Spec}D\subseteq X_A$ containing the point corresponding to $x$ over the closed point of $\mathrm{Spec}A$. By step 4.2, its base change $W_{A^{\mathrm{\prime }}}$ is an open of $\mathrm{Spec}{A}^{\mathrm{\prime }}$ containing its closed point, so $W_{A^{\mathrm{\prime }}}=\mathrm{Spec}{A}^{\mathrm{\prime }}$. Since $W_{A^{\mathrm{\prime }}}=\mathrm{Spec}(D{\otimes }_A{A}^{\mathrm{\prime }})$, the natural map $A^{\mathrm{\prime }}\to D{\otimes }_A{A}^{\mathrm{\prime }}$ is an isomorphism. Faithful flatness of $A\to A^{\mathrm{\prime }}$ kills the kernel and cokernel of $A\to D$, so $A\to D$ is an isomorphism. The complement $X_A\setminus W$ has empty base change to $A^{\mathrm{\prime }}$ by step 4.2; faithful flatness is surjective on spectra, so this complement is empty. Thus $j_A:X_A\to \mathrm{Spec}A$ is an isomorphism. In particular, the fibre of $j$ over $y$ has exactly one point and its stalk map ${\mathcal{O}}_{N,y}\to {\mathcal{O}}_{X,x}$ is an isomorphism. Since every point of $j(X)$ has such a unique preimage, step 4.1 makes $j$ a homeomorphism onto an open subset and these stalk isomorphisms identify its structure sheaf. Therefore $j$ is an open immersion.

By the open immersion of step 5.1, its image $J=j(X)$ is a quasi-compact open of $N$: $f$ is quasi-compact over the quasi-compact base by [F1]. Write using the filtered finite quasi-coherent subalgebras of [F5] and put $T_i={\mathrm{Spec}}_S{\mathcal{C}}_i$, so on affine base charts. On a finite affine cover of $S$, the quasi-compact open $J$ is a finite union of principal opens $D(c_1),\dots ,D(c_m)$ in an affine chart of $N$. The finitely many $c_a$ occur in one ${\mathcal{C}}_i$; hence $J$ is the inverse image of an open $J_i\subseteq T_i$ there. Because $S$ is quasi-separated, the pairwise intersections of the finite affine base cover are quasi-compact and admit finite affine refinements. On these finitely many overlap charts, equality of two such finite unions after passage to the filtered colimit is witnessed by finitely many radical-containment identities among their generators. Each identity holds at a later finite stage, since its finitely many coefficients and equalities occur there. After taking one common stage for the finite cover and its overlaps, the $J_i$ glue to a quasi-compact open $J_i\subseteq T_i$ with inverse image exactly $J$.

By step 5.1, where $J_k=J_i{\times }_{T_i}{T}_k$. The morphism $X\to S$ is of finite type by [F1]. Cover $J_i$ by finitely many affine opens $V=\mathrm{Spec}{A}_i$ lying over affine $\mathrm{Spec}R\subseteq S$. Because $N\to T_i$ is affine and $X\cong J$ is the inverse image of $J_i$, their inverse images in $X$ are affine, say $\mathrm{Spec}B$, with on the corresponding stages. Since $B$ is a finitely generated $R$-algebra, choose finitely many $R$-algebra generators of $B$ and lift each from some $A_k$. At a common later stage $A_k\to B$ is surjective. The finitely many affine charts therefore show, at one common stage, that $X\to J_k$ is a closed immersion. Let $Z\subseteq T_k$ be its scheme-theoretic image, defined by the kernel of ${\mathcal{O}}_{T_k}\to j_{k\ast }{\mathcal{O}}_X$. On $J_k$ the kernel cuts out exactly $X$, so $X=Z\cap J_k$ as schemes and $X\hookrightarrow Z$ is open. The closed subscheme $Z\to S$ is finite because $T_k\to S$ is finite. This proves the finite-stage factorization.

If $X=\mathrm{\varnothing }$, the pushforward algebra is the zero algebra and $N=\mathrm{\varnothing }$; the empty open immersion and finite zero-algebra stage satisfy the statement. Nonreduced and zero-ring affine charts are retained in step 1.1. AC is inherited through [F3]–[F5]; the local affine construction itself uses no additional choice. The open-immersion and finite-stage conclusions follow from steps 4.1–6.1.