Smooth proper curves, dominant morphisms and function fields

Part (1), injectivity of the pullback. Let $f:C\to D$ be a dominant $k$-morphism of smooth proper geometrically integral $k$-curves. Then $f({\eta }_C)={\eta }_D$ [F2], and the pullback $f^{\ast }:k(D)={\mathcal{O}}_{D,{\eta }_D}\to {\mathcal{O}}_{C,{\eta }_C}=k(C)$ is a homomorphism of fields [F1, F2]; being a map of fields into a nonzero field it is injective.

Part (1), an injective homomorphism gives a dominant rational map. Let $\phi :k(D)\hookrightarrow k(C)$ be an injective $k$-algebra homomorphism. Choose a finite affine cover $D={\bigcup }_i\mathrm{Spec}{A}_i$ (possible because $D$ is proper, hence quasi-compact [F1]). For each $i$, choose algebra generators $a_{i1},\dots ,a_{i{r}_i}$ of $A_i$. Choose a nonempty affine open $U_i=\mathrm{Spec}{R}_i\subset C$; each rational function $\phi (a_{ij})$ is a fraction in $\mathrm{Frac}(R_i)$, so after choosing a common nonzero denominator $g_i\in R_i$, all these images lie in the actual localization $(R_i{)}_{g_i}$. The induced $k$-algebra map $A_i\to (R_i{)}_{g_i}$ therefore defines a morphism $D(g_i)=\mathrm{Spec}((R_i{)}_{g_i})\to \mathrm{Spec}(A_i)\subseteq D$. These nonempty source opens are dense in the integral curve $C$. On a pairwise overlap $V$, the two maps agree at its generic point because they induce the same function-field map $\phi$. The equalizer is the pullback of the closed diagonal of the separated scheme $D$, so it is a closed subscheme of $V$. Its underlying closed subset contains the generic point, hence is all of the irreducible space $V$; since $V$ is reduced, [F4] makes the equalizer all of $V$. Thus the maps agree on overlaps and [F5] glues them on their union, a dense open subscheme of $C$, to a rational map $C⇢D$. Its induced map on function fields is $\phi$, which is injective, so the rational map is dominant by [F2].

Part (2), the affine normal model. Let $A\subseteq K$ be the $k$-subalgebra generated by a finite generating set of $K$ over $k$; then $A$ is a finite-type $k$-domain with $\mathrm{Frac}(A)=K$ and $\dim A={\mathrm{trdeg}}_{k}K=1$ [F1, F7]. Let $B\subseteq K$ be the integral closure of $A$ in $K=\mathrm{Frac}(A)$. It is finite over $A$ [F6] and thus a finite-type $k$-algebra [F26]. By [F16], $A$ is Noetherian; [F17] makes $B$ Noetherian, and [F18] makes it integrally closed. The normality-locality criterion [F20] and definition [F19] therefore make $B$ a normal Noetherian domain. Also $\mathrm{Frac}(B)=K$, so [F7] gives $\dim B=1$. Thus $\mathrm{Spec}B$ is an integral, dimension-one affine $k$-scheme; geometric integrality is not asserted at this stage.

Part (2), geometric integrality of the function field. Fix a finite subextension $l\mathrm{/}k$ of $\bar{k}\mathrm{/}k$. By [F31], write $l=k(\alpha )$ with irreducible minimal polynomial $m\in k[T]$. If $m$ factored over $K$, take monic factors in $K[T]$ and split $m$ in an algebraic closure of $K$. Every coefficient of either factor is a symmetric expression in roots algebraic over $k$, hence is algebraic over $k$. Relative algebraic closedness forces these coefficients into $k$, contradicting irreducibility of $m$ over $k$. Therefore $K{\otimes }_{k}l\cong K[T]\mathrm{/}(m)$ is a field. The inclusions for finite subextensions remain injective by [F31]. Every finite family of elements of $K{\otimes }_k\bar{k}$ belongs to one such tensor product, since its finitely many coefficients generate a finite subextension. Thus $K{\otimes }_k\bar{k}$ is a domain.

Part (1), extension and uniqueness. By [F3] and Choice [F15] the rational map of step 1.2 extends to a unique morphism $F:C\to D$; its pullback is $\phi$. If $f_1,f_2:C\to D$ have the same pullback on function fields, they agree at the generic point. Because $D$ is separated, their equalizer is a closed subscheme of $C$; its underlying closed subset contains the generic point and therefore is all of the irreducible space $C$. Since $C$ is reduced, a closed subscheme with the same underlying space is $C$ itself [F4], so $f_1=f_2$.

Part (2), projective closure. Starting from the affine normal model constructed in step 1.3, choose $k$-algebra generators $b_1,\dots ,b_m$ of $B$ and let $I=\ker (k[y_1,\dots ,y_m]\twoheadrightarrow B)$; this gives a closed immersion $\mathrm{Spec}B\hookrightarrow {\mathbf{A}}_k^m$ [F8]. Put $S=k[x_0,\dots ,x_m]$, identify $k[y_1,\dots ,y_m]$ with the degree-zero subring of $S[x_0^{-1}]$ by $y_i\mapsto x_i\mathrm{/}{x}_0$, and define the homogeneous ideal $J=(I\cdot S[x_0^{-1}])\cap S\mathrm{.}$ Because $I$ is prime, its extension to $S[x_0^{-1}]$ is prime, and its contraction $J$ is homogeneous and prime; moreover $x_0\mathrm{\notin }J$. Thus $\bar{X}=\mathrm{Proj}(S\mathrm{/}J)$ is an integral closed subscheme of ${\mathbb{P}}_k^m$. Its standard chart $D_{+}(x_0)$ has coordinate ring $k[y_1,\dots ,y_m]\mathrm{/}I=B$, so $\mathrm{Spec}B$ is an open dense subscheme of $\bar{X}$, with function field $K$. Every nonempty affine open of $\bar{X}$ is an integral finite-type $k$-scheme with function field $K$, so [F7] gives dimension one. Since $\bar{X}$ is closed in projective space, it is proper over $k$ by [F9]. At this point we use only that $\bar{X}$ is an integral, separated, finite-type dimension-one $k$-scheme; geometric integrality is established next.

Part (2), the projective closure is a curve. Each nonempty affine chart $\mathrm{Spec}R$ of $\bar{X}$ embeds its coordinate ring in $K$. By [F31] this gives an injection $R{\otimes }_k\bar{k}\hookrightarrow K{\otimes }_k\bar{k}$, whose target is a domain by step 1.4. These nonzero domains are the charts of ${\bar{X}}_{\bar{k}}$ by [F25]. Any two charts meet after base change: their original nonempty intersection contains a nonempty affine open, whose coordinate algebra also remains a nonzero domain after tensoring. The charts are irreducible and have nonempty open intersections, so their union is irreducible; it is also reduced and nonempty. Hence $\bar{X}$ is geometrically integral. Together with step 2.2, this makes it a curve in [F1], so the hypothesis of [F10] is satisfied.

Part (1), bijectivity. By steps 1.1 and 2.1 the assignment $f\mapsto f^{\ast }$ is a well-defined map from dominant $k$-morphisms $C\to D$ to injective $k$-algebra homomorphisms $k(D)\hookrightarrow k(C)$; it is injective by step 2.1 and surjective by steps 1.2 and 2.1, hence a bijection onto its image, which is the set of all injective homomorphisms by step 1.2. This proves (1).

Part (2), normalization. Let $\nu :X\to \bar{X}$ be the normalization [F10], applied to the geometrically integral curve established in steps 2.2 and 3.1. Then $X$ is integral and normal, $\nu$ is finite and birational, and $k(X)=K$. Since $\bar{X}$ is proper over $k$ and $\nu$ is finite, the composite $X\to \bar{X}\to \mathrm{Spec}k$ is proper by [F9].

Part (2), dimension and smoothness. The finite morphism $X\to \bar{X}$ and the finite standard affine cover of the projective scheme $\bar{X}$ give a finite affine cover of $X$ by [F28]; on each chart its ring is module-finite over a finite-type $k$-algebra, hence is finite type over $k$ by [F26]. Thus $X$ is finite type. For every nonempty affine open $V=\mathrm{Spec}R\subseteq X$, [F1] identifies $\mathrm{Frac}(R)$ with $k(X)=K$; [F7] then gives $\dim R={\mathrm{trdeg}}_{k}K=1$. This gives chain dimension one on $X$: any chain in $X$ restricts to an affine chart containing the generic point of its smallest member, and the strict inclusions persist after restriction; conversely each affine chart is open, so its chains give chains in $X$ by taking closures. If $x$ is not the generic point, choose an affine neighborhood $V=\mathrm{Spec}R$ and let $\mathfrak{p}$ correspond to $x$. Then $\mathfrak{p}\mathrm{\ne }(0)$, so $R_{\mathfrak{p}}$ has dimension at least one, while the chain-dimension bound makes it at most one. The local ring is Noetherian because $R$ is finite type over the field [F16], and integrally closed because $X$ is normal. It is therefore a one-dimensional Noetherian local integrally closed domain, hence a DVR by [F11], and thus regular. The generic local ring is the field $k(X)$ [F1], regular of dimension zero. So all local rings of $X$ are regular; as $X$ is finite type over perfect $k$, [F12] makes $X\to \mathrm{Spec}k$ smooth. This argument does not call $X$ a curve before geometric integrality is proved.

Part (2), geometric integrality of the normalization. Every nonempty affine coordinate ring $R$ of $X$ embeds in its function field $K$ [F1]. As in step 3.1, [F25] and [F31] identify its base-changed chart with the spectrum of the nonzero domain $R{\otimes }_k\bar{k}\subseteq K{\otimes }_k\bar{k}$. Nonempty intersections remain nonempty by the same argument applied to an affine open in the intersection. Thus $X_{\bar{k}}$ is nonempty, reduced and irreducible, and $X$ is geometrically integral by [F27].

Part (2), existence and uniqueness. Steps 2.2–5.2 establish that $X$ is a smooth proper geometrically integral curve, and along the construction $k(X)=K$, giving a model $(X,\phi )$ with $\phi :K\to k(X)$ the identity identification. For uniqueness let $(C,\phi )$ and $(C^{\mathrm{\prime }},{\phi }^{\mathrm{\prime }})$ be two models. Then $\alpha :={\phi }^{\mathrm{\prime }}\circ {\phi }^{-1}:k(C)\to k(C^{\mathrm{\prime }})$ is an isomorphism. By part (1), it determines a unique dominant morphism $\psi :C^{\mathrm{\prime }}\to C$ whose pullback is ${\psi }^{\ast }=\alpha$; thus ${\phi }^{\mathrm{\prime }}={\psi }^{\ast }\circ \phi$. Applying part (1) to ${\alpha }^{-1}$ gives ${\psi }^{\mathrm{\prime }}:C\to C^{\mathrm{\prime }}$ with $({\psi }^{\mathrm{\prime }}{)}^{\ast }={\alpha }^{-1}$. The pullback of ${\psi }^{\mathrm{\prime }}\circ \psi :C^{\mathrm{\prime }}\to C^{\mathrm{\prime }}$ is ${\psi }^{\ast }\circ ({\psi }^{\mathrm{\prime }}{)}^{\ast }=\alpha \circ {\alpha }^{-1}={\mathrm{i}\mathrm{d}}_{k(C^{\mathrm{\prime }})}$, so uniqueness in part (1) gives ${\psi }^{\mathrm{\prime }}\circ \psi ={\mathrm{i}\mathrm{d}}_{C^{\mathrm{\prime }}}$; similarly $\psi \circ {\psi }^{\mathrm{\prime }}={\mathrm{i}\mathrm{d}}_C$. Hence $\psi$ is the unique $k$-isomorphism $C^{\mathrm{\prime }}\to C$ satisfying the stated relation.

Conclusion and object correspondence. Steps 1.3–6.1 prove that every field object in (2) has a smooth proper geometrically integral curve model, unique up to the stated unique isomorphism; step 3.2 proves full faithfulness for every field $k$. Conversely, for any smooth proper geometrically integral curve $C$ over $k$, its function field $K=k(C)$ is finitely generated over $k$ by [F1], has transcendence degree one by [F29], and has $k$ relatively algebraically closed by [F30]. Thus its function field is an object of the stated field category. The object assignments and the contravariant bijection of morphisms from step 3.2 give the claimed equivalence. The Axiom of Choice is inherited from the extension, normalization, properness, and geometric-integrality suppliers cited at their uses.