Normalization of an integral finite-type curve by gluing affine integral closures

Delete empty members of the finite cover. Each remaining $A_i$ is a domain, since $U_i$ is a nonempty affine open of the integral scheme $C$ [F15], and $A_i$ is a finitely generated $k$-algebra because $C$ is of finite type over $k$. By [F4] the fraction field $\mathrm{Frac}(A_i)$ is canonically identified with $K={\mathcal{O}}_{C,\eta }$, and these identifications are compatible on overlaps, so all of them may be regarded as subfields of one copy of $K$. Each $U_i\cap U_j$ is affine by [F5], since $C$ is separated over $\mathrm{Spec}k$ and all affine opens of $C$ lie over the single affine open $\mathrm{Spec}k$.

Let $B_i\subseteq K$ be the integral closure of $A_i$ in $K$ [F9]. By [F1], $B_i$ is a finite $A_i$-module and a domain with fraction field $K$. The integral-closure theorem [F17] makes $B_i$ integrally closed. Since $k$ is a principal ideal domain, [F8] makes each finite-type $A_i$ Noetherian; [F19] then makes the module-finite $A_i$-algebra $B_i$ Noetherian. Applying both directions of [F18] to $B_i$, all of its prime localizations are integrally closed, so $B_i$ is normal in the Noetherian-ring sense [F10].

For $0\mathrm{\ne }f\in A_i$ the integral closure of the principal localisation $(A_i{)}_f$ in $K$ is exactly $(B_i{)}_f$ and $(B_i{)}_f$ is a finite $(A_i{)}_f$-module, by [F2]. Consequently the ring $B_i$ attached to the chart is determined on each principal open $D(f)\subseteq U_i$ by that open alone, namely as $(B_i{)}_f$ inside $K$.

Gluing data. Fix $i,j$ and put $W=U_i\cap U_j$, which is affine by [F5]. Let $V_{ij}$ be the inverse image of $W$ in $\mathrm{Spec}(B_i)$. We construct compatible identifications locally on $W$. For each point $w\in W$, choose principal opens $D(f)\subseteq U_i$ and $D(g)\subseteq U_j$ containing $w$ and contained in $W$; such choices exist by [F13]. On $D(f)$ the restriction of $g$ is a regular function, hence is represented by an element of $(A_i{)}_f$; write it as $c\mathrm{/}{f}^r$. Then $D(f)\cap D(g)=D(fc)$ as an open of $U_i$. Similarly, on $D(g)$ the restriction of $f$ is represented by $d\mathrm{/}{g}^s$ in $(A_j{)}_g$, so the same intersection is $D(gd)$ as an open of $U_j$. Thus $D(fc)=D(gd)$ is a common principal-open neighborhood of $w$ in $W$. Its coordinate rings, computed in either chart, are the same subring of $K$, since both are $\mathrm{\Gamma }(D(fc),{\mathcal{O}}_C)$. By [F2], the integral closures of this ring in $K$ are respectively $(B_i{)}_{fc}$ and $(B_j{)}_{gd}$, so these localizations are equal inside $K$. The identity of that ring induces an isomorphism between the corresponding opens in the two normalization charts. These common opens cover $W$; the isomorphisms agree on further intersections because every ring map is the identity inside $K$. They therefore glue to an isomorphism $V_{ij}\to V_{ji}$ over $W$. The same identity-in-$K$ argument gives inverse maps and the cocycle condition on triple overlaps. No single distinguished open of $U_i$ is assumed to be represented by one element of $A_j$.

Gluing. By [F3] the affine schemes $\mathrm{Spec}(B_i)$, equipped with the open subschemes $V_{ij}$ and the compatible isomorphisms ${\phi }_{ij}$, glue to a scheme $C^{\mathrm{n}\mathrm{u}}$ on which the charts $\mathrm{Spec}(B_i)$ form an open affine cover.

The morphism $\nu$. Each inclusion $A_i\subseteq B_i$ induces a $k$-morphism $\mathrm{Spec}(B_i)\to U_i$ by [F12]. On the common principal-open refinements from step 1.4, the chart isomorphism is induced by the identity of the localized integral-closure ring inside $K$; both composites to $C$ are therefore the same map to the overlap $W$. The chart morphisms agree on the open overlaps and glue by [F6] to $\nu :C^{\mathrm{n}\mathrm{u}}\to C$. The transition maps are these normalization-chart isomorphisms over $W$, not inclusions of one chart into the other.

$C^{\mathrm{n}\mathrm{u}}$ is integral, normal and has function field $K$. Every chart $\mathrm{Spec}(B_i)$ is integral and has generic point with local ring $K$. Any two remaining $U_i,U_j$ meet in a nonempty open because $C$ is irreducible. The inverse image of that overlap contains the generic point of each normalization chart, and the transition maps identify those generic points by the identity of $K$. They therefore give one point $\eta$ lying in every chart. It is dense in each chart because each $B_i$ is a domain, hence dense in $C^{\mathrm{n}\mathrm{u}}$; this proves global irreducibility. The charts are reduced, so the glued scheme is reduced and therefore integral. Their finite affine cover has Noetherian coordinate rings by step 1.2, so $C^{\mathrm{n}\mathrm{u}}$ is Noetherian; [F18] makes every local ring integrally closed, hence the scheme is normal by [F20]. The common generic local ring is $K$, so $k(C^{\mathrm{n}\mathrm{u}})=K$.

Finiteness. For each $i$ we have ${\nu }^{-1}(U_i)=\mathrm{Spec}(B_i)$ by construction, and $B_i$ is a finite $A_i$-module by [F1]. The cover $C={\bigcup }_i{U}_i$ is a finite affine open cover of the target, so the local criterion [F7] shows that $\nu$ is finite; in particular $\nu$ is affine by the choice-free first half of [F7].

Birationality. The map $\nu$ sends the common generic point $\eta$ of step 2.3 to the generic point of $C$ and induces the identity map $K\to K$ on function fields. It is therefore dominant and birational by [F11].

Initiality. Let $h:Z\to C$ be finite and birational, with $Z$ normal and integral. For each $i$, $h^{-1}(U_i)=\mathrm{Spec}(D_i)$ and $D_i$ is a finite $A_i$-algebra by [F7, F14]. The preimage contains the generic point, so $D_i$ is a domain. Birationality and [F4] identify $\mathrm{Frac}(D_i)$ with $K$; under these identifications the map $A_i\to D_i$ is injective, so regard it as an inclusion. The finite affine covers and [F8], [F19] make $C$ and $Z$ Noetherian; normality of $Z$ says every localization $(D_i{)}_{\mathfrak{q}}$ is an integrally closed domain [F20]. By the converse direction of [F18], $D_i$ itself is integrally closed in $K$. The finite-morphism theorem [F21] makes every element of $D_i$ integral over $A_i$, so $D_i\subseteq B_i$. Conversely, each $b\in B_i$ is integral over $A_i\subseteq D_i$ and lies in $K$, so integral closedness of $D_i$ gives $b\in D_i$. Thus $D_i=B_i$ as subrings of $K$ for every $i$. The identity ring maps on these equal chart algebras induce chart isomorphisms in both directions, and they glue by [F6] to morphisms $\varphi :C^{\mathrm{n}\mathrm{u}}\to Z$ and $\psi :Z\to C^{\mathrm{n}\mathrm{u}}$ over $C$. They are inverse because their restrictions on each affine chart are identities. Any morphism $C^{\mathrm{n}\mathrm{u}}\to Z$ over $C$ induces the identity on the generic function field $K$; its chart ring maps are therefore the identity on $D_i=B_i\subseteq K$, so it equals $\varphi$. The same argument for a morphism $Z\to C^{\mathrm{n}\mathrm{u}}$ makes it equal to $\psi$. Thus both maps are unique, and in particular $\varphi$ proves initiality in the stated direction.

Uniqueness of the normalization. Let $(C^{\mathrm{\prime }},{\nu }^{\mathrm{\prime }})$ be another normal integral scheme finite and birational over $C$. Step 4.1 gives unique maps $u:C^{\mathrm{n}\mathrm{u}}\to C^{\mathrm{\prime }}$ and $v:C^{\mathrm{\prime }}\to C^{\mathrm{n}\mathrm{u}}$ over $C$. Uniqueness forces $v\circ u$ and $u\circ v$ to be the identity maps. Hence $u$ and $v$ are inverse isomorphisms, unique over $C$.

Conclusion and Choice accounting. Steps 1.2 and 1.3 prove finiteness of the affine integral closures and compatibility with principal localization; steps 1.4, 2.1 and 2.2 construct the glued scheme and morphism; steps 3.1 and 3.2 prove that the morphism is finite, affine and birational; step 2.3 proves integrality, normality and the function-field identity; and steps 4.1 and 5.1 prove initiality and uniqueness. The Axiom of Choice is inherited through the normality-locality criterion [F18], the local criterion for finiteness [F7], and the finite-morphism integrality theorem [F21]; these are used in steps 1.2, 3.1, and 4.1.