Chow lemma for proper Noetherian schemes

If $X=\mathrm{\varnothing }$ take $X^{\mathrm{\prime }}=U=\mathrm{\varnothing }$, $N=0$, and both maps empty; all assertions hold, the immersion being the identity of the empty scheme. Assume $X\mathrm{\ne }\mathrm{\varnothing }$. By [F5] the scheme $X$ is Noetherian and quasi-compact with finitely many irreducible components $X_1,\dots ,X_r$, $r\ge 1$, and generic points ${\eta }_i$.

For every point $x\in X$ choose by [F5] an affine open $U_x$ containing $x$ and all generic points ${\eta }_1,\dots ,{\eta }_r$. Since $X$ is quasi-compact, finitely many of them, say $U_1,\dots ,U_m$, cover $X$, and each $U_i$ contains every ${\eta }_j$.

The open subscheme $U:=U_1\cap \cdots \cap U_m$ is dense and nonempty: it contains ${\eta }_1,\dots ,{\eta }_r$, and every irreducible component $\overline{\{{\eta }_j\}}$ meets $U$, so the closure of $U$ contains each component and hence equals $X$.

Replace $X$ by the schematic closure $X^{\ast }$ of $U$ in $X$: by [F4] applied to the open immersion $U\hookrightarrow X$ (quasi-compact because $X$ is Noetherian) there is a closed immersion $X^{\ast }\to X$ which is a surjective closed immersion, restricts to an isomorphism over $U$, and makes $U$ schematically dense in $X^{\ast }$; also $X^{\ast }$ is Noetherian and $U_i^{\ast }:=X^{\ast }\cap U_i$ is affine and contains $U$, the finitely many $U_i^{\ast }$ covering $X^{\ast }$. Since properness, surjectivity and the isomorphism over $U$ are preserved by composing with the closed immersion $X^{\ast }\to X$, and since immersions into ${\mathbb{P}}_S^N$ compose with that closed immersion, it suffices to prove the lemma for $X^{\ast }$; hence from now on we assume $U$ is schematically dense in $X$, replacing $X,U_i$ by $X^{\ast },U_i^{\ast }$.

For each $i$, the affine open $U_i$ is locally of finite type over $S$ by restriction of $f$. Apply [F6] directly to obtain an $S$-immersion $j_i:U_i\to {\mathbb{P}}_S^{n_i}$ for some $n_i\ge 0$. This uses no assertion that $U_i$ maps into a single affine open of $S$.

Let $Z_i\subseteq {\mathbb{P}}_S^{n_i}$ be the scheme-theoretic image of $j_i$, which exists by [F3] because $U_i$ is Noetherian and quasi-compact. By [F6], write $j_i$ as a closed immersion $U_i\hookrightarrow W_i$ followed by an open immersion $W_i\hookrightarrow {\mathbb{P}}_S^{n_i}$. The restriction clause of [F3] identifies $Z_i\cap W_i$ with the scheme-theoretic image of that closed immersion, namely $U_i$. Thus $U_i\to Z_i$ is an open immersion, schematically dense by [F3], and $Z_i\to S$ is proper as a closed subscheme of the proper $S$-scheme ${\mathbb{P}}_S^{n_i}$.

Let $P:={\mathbb{P}}_S^{n_1}{\times }_S\cdots {\times }_S{\mathbb{P}}_S^{n_m}$ with projections ${\mathrm{p}\mathrm{r}}_i$, and let $j:U\to P$ be $(j_1{\mathrm{\mid }}_U,\dots ,j_m{\mathrm{\mid }}_U)$. The map $U\to U_1{\times }_S\cdots {\times }_S{U}_m$ is a closed immersion: it is the base change of the closed multi-diagonal $X\to X_S^m$ of separated $X\to S$, since $U=U_1\cap \cdots \cap U_m$. For the opens $W_i$ of 1.6, the product of the closed immersions $U_i\hookrightarrow W_i$ is a closed immersion ${\prod }_S{U}_i\hookrightarrow {\prod }_S{W}_i$, and ${\prod }_S{W}_i$ is open in $P$. Hence $j$ is a closed immersion into that open product and thus an immersion into $P$. Its source is Noetherian, so the scheme-theoretic image $Z\subseteq P$ exists by [F3]. Restricting to the open product ${\prod }_S{W}_i$ gives exactly $j(U)$, so $U\to Z$ is an open immersion and $U$ is schematically dense in $Z$. The morphism $Z\to S$ is proper: the product $P\to S$ is proper by successive base change and composition of the projective-space maps [F1,F2], and $Z\hookrightarrow P$ is a closed immersion.

For each $i$ the projection ${\mathrm{p}\mathrm{r}}_i{\mathrm{\mid }}_Z:Z\to {\mathbb{P}}_S^{n_i}$ factors through $Z_i$: the closed subscheme ${\mathrm{p}\mathrm{r}}_i^{-1}(Z_i)\subseteq P$ is a closed subscheme through which $j$ factors, because ${\mathrm{p}\mathrm{r}}_i\circ j=j_i{\mathrm{\mid }}_U$ factors through $Z_i$; by minimality of $Z$ among closed subschemes of $P$ through which $j$ factors we have $Z\subseteq {\mathrm{p}\mathrm{r}}_i^{-1}(Z_i)$, and hence ${\mathrm{p}\mathrm{r}}_i{\mathrm{\mid }}_Z$ factors through the projection ${\mathrm{p}\mathrm{r}}_i^{-1}(Z_i)\to Z_i$ of the fibre product. Denote the induced morphism by $p_i:Z\to Z_i$; it is proper because $Z$ and $Z_i$ are proper over $S$ and $Z_i$ is separated over $S$ (as a closed subscheme of the separated $S$-scheme ${\mathbb{P}}_S^{n_i}$), using [F2].

Let $V_i:=p_i^{-1}(U_i)\subseteq Z$; this is an open subscheme, and $p_i{\mathrm{\mid }}_{V_i}:V_i\to U_i$ is proper, being the base change of the proper morphism $p_i$ along the open immersion $U_i\hookrightarrow Z_i$; it is surjective because its image is closed in $U_i$ (proper morphisms have closed image) and contains $p_i(U\cap V_i)=U$, which is dense in $U_i$. Set $X^{\mathrm{\prime }}:=V_1\cup \cdots \cup V_m\subseteq Z$; this is an open subscheme, so $X^{\mathrm{\prime }}\to Z$ is an open immersion, and composing with the closed immersion $Z\hookrightarrow P$ gives an immersion $X^{\mathrm{\prime }}\to P$. By the iterated Segre embedding [F7] the product $P$ is a closed subscheme of ${\mathbb{P}}_S^N$ for $N=(n_1+1)\cdots (n_m+1)-1$, so the composite $X^{\mathrm{\prime }}\to {\mathbb{P}}_S^N$ is an immersion over $S$ (composition of an open immersion, a closed immersion and a closed immersion), which is the required $\iota$.

The morphisms $p_i{\mathrm{\mid }}_{V_i}:V_i\to U_i\hookrightarrow X$ glue to a morphism $\pi :X^{\mathrm{\prime }}\to X$: on $V_i\cap V_j$ the two composites agree after restriction to the schematically dense open $U$ (both equal the identity of $U$), and they agree on all of $V_i\cap V_j$ because their difference, viewed through the closed diagonal ${\mathrm{\Delta }}_{X\mathrm{/}S}\subseteq X{\times }_{S}X$ of the separated $S$-scheme $X$, has closed preimage in $V_i\cap V_j$ containing the schematically dense open $U$, forcing equality; here $U$ is schematically dense in the open subscheme $V_i\cap V_j$ because schematic density is checked by the vanishing of a kernel sheaf and passes to open subschemes.

For each $i$, ${\pi }^{-1}(U_i)=V_i$. The inclusion $V_i\subseteq {\pi }^{-1}(U_i)$ is part of the construction. Conversely, cover ${\pi }^{-1}(U_i)$ by the opens $W_{ij}:=V_j\cap {\pi }^{-1}(U_i)$. Both $p_i{\mathrm{\mid }}_{W_{ij}}$ and $j_i\circ \pi {\mathrm{\mid }}_{W_{ij}}$ map $W_{ij}$ to the separated $S$-scheme $Z_i$ and agree on $U\subseteq W_{ij}$. The open $U$ is schematically dense in $W_{ij}$ because it is schematically dense in $Z$ and $W_{ij}$ is open; separatedness and [F4] force the two maps to agree everywhere. Hence $p_i(W_{ij})\subseteq j_i(U_i)=U_i$, so $W_{ij}\subseteq p_i^{-1}(U_i)=V_i$. Their union is ${\pi }^{-1}(U_i)$, proving equality.

The morphism $\pi$ is proper: by 1.11 the restrictions $\pi {\mathrm{\mid }}_{{\pi }^{-1}(U_i)}:{\pi }^{-1}(U_i)\to U_i$ are identified with the proper morphisms $p_i{\mathrm{\mid }}_{V_i}$, and properness is local on the target for the cover $X=U_1\cup \cdots \cup U_m$.

The morphism $\pi$ is surjective: its image is closed in $X$ (properness) and contains $\pi (V_i)=p_i(V_i)=U_i$ for every $i$ by 1.9, hence equals $X$.

Let $E:={\pi }^{-1}(U)$, an open subscheme of $Z$ containing the schematically dense open $U$. The two $S$-morphisms $E\to P$ given by the inclusion $E\hookrightarrow Z\hookrightarrow P$ and by $j\circ \pi {\mathrm{\mid }}_E$ agree on $U$, where $\pi$ is the identity. Since $P$ is separated over $S$, [F4] makes them equal on $E$. The closed immersion $Z\hookrightarrow P$ is a monomorphism, so as maps into $Z$ this says $j_Z\circ \pi {\mathrm{\mid }}_E={\mathrm{id}}_E$, where $j_Z:U\hookrightarrow Z$ is the open immersion from 1.7. Also $\pi \circ j_Z={\mathrm{id}}_U$ by construction. Therefore $E=j_Z(U)$ and ${\pi }^{-1}(U)\to U$ is an isomorphism.

If $f$ is proper, then $X^{\mathrm{\prime }}$ is proper over $S$, being the composite of the proper morphism $\pi :X^{\mathrm{\prime }}\to X$ of 1.12 with $f$; the immersion $\iota :X^{\mathrm{\prime }}\to {\mathbb{P}}_S^N$ is a morphism of $S$-schemes with $X^{\mathrm{\prime }}$ proper over $S$ and ${\mathbb{P}}_S^N$ separated over $S$, hence $\iota$ is proper, in particular its image is closed in ${\mathbb{P}}_S^N$; by [F2] the immersion $\iota$ is then a closed immersion, and $X^{\mathrm{\prime }}$ is projective over $S$ in the sense of Projective morphisms before Proj.

Boundary and choice accounting. The empty case is 1.1; $m=1$ (one affine chart) and $r=1$ (one irreducible component) are included in the arguments above, and $N=0$ is allowed when all $n_i=0$ (then $P={\mathbb{P}}_S^0$ and the Segre embedding is the identity). The Axiom of Choice is a hypothesis of the published scheme-image and closed-subscheme correspondence used in [F3], and licenses the finite affine and principal-open selections in steps 1.1–1.8. Later steps use only finite selections from those covers.