Closed gluing of two projective three-spaces is proper

Put $Z:=Z_1$, so that $Z$ is a $k$-scheme and $j_1=z_1:Z\to X_1$ and $j_2=z_2\sigma :Z\to X_2$ are morphisms of $k$-schemes. Both are closed immersions: $j_1$ by the data, and for $j_2$ the underlying map is the composite of the homeomorphism $\sigma$ onto $\mathrm{\mid }{Z}_2\mathrm{\mid }$ with the homeomorphism $z_2$ onto the closed subset $\mathrm{\mid }{Z}_2\mathrm{\mid }$, hence a homeomorphism onto a closed subset, while the sheaf map ${\mathcal{O}}_{X_2}\to (z_2\sigma {)}_{\ast }{\mathcal{O}}_Z$ equals ${\mathcal{O}}_{X_2}\to z_{2\ast }{\mathcal{O}}_{Z_2}$ under the identification ${\sigma }_{\ast }{\mathcal{O}}_Z={\mathcal{O}}_{Z_2}$, hence is surjective.

Apply [F4] with $S=\mathrm{Spec}k$ to the closed immersions $j_1$ and $j_2$. The pushout $T=X_1{⨿}_Z{X}_2$ exists as a $k$-scheme with structure morphisms $a:X_1\to T$ and $b:X_2\to T$, and $a$ and $b$ are closed immersions with $\mathrm{\mid }T\mathrm{\mid }=\mathrm{\mid }{X}_1\mathrm{\mid }\cup \mathrm{\mid }{X}_2\mathrm{\mid }$ and $\mathrm{\mid }{X}_1\mathrm{\mid }\cap \mathrm{\mid }{X}_2\mathrm{\mid }=\mathrm{\mid }Z\mathrm{\mid }$; the square $Z\to X_1$, $Z\to X_2$, $X_1\to T$, $X_2\to T$ is Cartesian; every point of $Z$ has an open neighbourhood in $T$ of the form $\mathrm{Spec}(A{\times }_{C}B)$ with $A=\mathrm{\Gamma }(U,\mathcal{O})$, $B=\mathrm{\Gamma }(V,\mathcal{O})$, $C=\mathrm{\Gamma }(j_1^{-1}(U),\mathcal{O})$ for affine opens $U\subseteq X_1$, $V\subseteq X_2$ with $j_1^{-1}(U)=j_2^{-1}(V)$, while the points of $T$ outside $Z$ lie in the open subschemes $X_1\setminus Z$ and $X_2\setminus Z$. By [F5] the closed immersions $a,b$ exhibit $X_1,X_2$ as closed subschemes of $T$; this is the existence clause of the statement.

Each $X_i$ is the relative projective space ${\mathbb{P}}_k^3$ of [F2], so by [F3] with $S=\mathrm{Spec}k$ and $n=3$ each structure morphism $X_i\to \mathrm{Spec}k$ is proper; by [F7] each $X_i\to \mathrm{Spec}k$ is therefore separated, of finite type and universally closed, separated meaning that its diagonal is a closed immersion by [F8], and by [F9] each $X_i\to \mathrm{Spec}k$ is in particular quasi-compact and locally of finite type.

Let $U\subseteq T$ be an affine open. By [F6] applied to the closed immersion $a:X_1\to T$ and the affine open $U$, the preimage $a^{-1}(U)$ is the affine scheme $\mathrm{Spec}(\mathrm{\Gamma }(U,{\mathcal{O}}_T)\mathrm{/}I)$ for an ideal $I$, and it coincides with the open subscheme $U\cap X_1$ of $X_1$. The same holds for $b^{-1}(U)=U\cap X_2$ inside $X_2$.

Let $R\subseteq K$ be a valuation ring with fraction field $K$ and let $\eta$ be the generic point of $\mathrm{Spec}R$. Since $R$ is a subring of the field $K$ by [F24] and $K$ is its field of fractions [F26], $R$ is a domain, so $(0)$ is a prime ideal, $V((0))=\mathrm{Spec}R$ by [F20], and by [F21] this set is closed; hence by [F27] the point $\eta =(0)$ is the generic point of $\mathrm{Spec}R$ and $\mathrm{Spec}R=\overline{\{\eta \}}$. Moreover, if $O\subseteq \mathrm{Spec}R$ is an open subset containing the point $\mathfrak{m}$ corresponding to the unique maximal ideal of $R$ [F25], then $O$ is the complement of a closed set $V(I)$ for some ideal $I$ by [F20] and [F21]; $\mathfrak{m}\mathrm{\notin }V(I)$ means $I⊈\mathfrak{m}$, so $I$ contains an element outside $\mathfrak{m}$, which is a unit of $R$ by [F25], whence $I=R$, $V(I)=V(R)=\mathrm{\varnothing }$ and $O=\mathrm{Spec}R$. So every open subset of $\mathrm{Spec}R$ containing $\mathfrak{m}$ is all of $\mathrm{Spec}R$.

Since $X_i\to \mathrm{Spec}k$ is quasi-compact and $\mathrm{Spec}k$ is quasi-compact and affine, [F10] and [F31] show that $X_i=(X_i\to \mathrm{Spec}k{)}^{-1}(\mathrm{Spec}k)$ is quasi-compact. The closed immersions $a,b$ are homeomorphisms onto the closed subsets $\mathrm{\mid }{X}_1\mathrm{\mid },\mathrm{\mid }{X}_2\mathrm{\mid }\subseteq \mathrm{\mid }T\mathrm{\mid }$, so these are quasi-compact subspaces of $\mathrm{\mid }T\mathrm{\mid }$ homeomorphic to $X_1,X_2$, and $\mathrm{\mid }T\mathrm{\mid }=\mathrm{\mid }{X}_1\mathrm{\mid }\cup \mathrm{\mid }{X}_2\mathrm{\mid }$. A union of two quasi-compact subspaces is quasi-compact: given an open cover of the union, intersecting its members with each of the two subspaces gives open covers of the subspaces, from which finitely many members can be selected by [F30], and the finitely many selected members cover the union. Hence $T$ is quasi-compact by [F11], and $T\to \mathrm{Spec}k$ is quasi-compact by [F31] applied to the affine cover $\{\mathrm{Spec}k\}$.

Let $z\in Z$ and let $\mathrm{Spec}(A{\times }_{C}B)$ be the chart of [F4] part (3) around $z$, so that $A=\mathrm{\Gamma }(U,{\mathcal{O}}_{X_1})$ and $B=\mathrm{\Gamma }(V,{\mathcal{O}}_{X_2})$ for affine opens $U\subseteq X_1$, $V\subseteq X_2$ with $j_1^{-1}(U)=j_2^{-1}(V)$ and $C=\mathrm{\Gamma }(j_1^{-1}(U),{\mathcal{O}}_Z)$. By [F6] applied to the closed immersions $j_1,j_2$ over the affine opens $U,V$ the maps $A\to C$ and $B\to C$ are surjective with $j_1^{-1}(U)=\mathrm{Spec}(A\mathrm{/}I)$, $C=A\mathrm{/}I$ for $I=\ker (A\to C)$ and similarly $C=B\mathrm{/}J$ for $J=\ker (B\to C)$. Since $X_1\to \mathrm{Spec}k$ is locally of finite type by step 1.3, the affine-locality [F12] makes $A$ a finitely generated $k$-algebra, and likewise $B$; then $C$ is a finitely generated $k$-algebra, being a quotient of $A$. Choose finitely many $k$-algebra generators $c_1,\dots ,c_r$ of $C$ and lifts ${\alpha }_{\mathrm{\ell }}\in A$, ${\beta }_{\mathrm{\ell }}\in B$ of them, and finite generating lists $i_1,\dots ,i_s$ of $I$ and $j_1,\dots ,j_t$ of $J$: such lists exist because by [F32] $A$ is a quotient of a polynomial ring over the field $k$, ideals of $A$ are images of ideals of that polynomial ring, and those have finite generating lists by [F33]. Also choose finite $k$-algebra generating lists $a_1,\dots ,a_u$ of $A$ and $b_1,\dots ,b_v$ of $B$. Surjectivity onto $C$ gives lifts ${\widehat{b}}_p\in B$ of the image of $a_p$, and ${\widehat{a}}_q\in A$ of the image of $b_q$. Let $E$ be the $k$-subalgebra of $A{\times }_{C}B$ generated by the finitely many pairs $({\alpha }_{\mathrm{\ell }},{\beta }_{\mathrm{\ell }})$, $(i_{\mu },0)$, $(0,j_{\nu })$, $(a_p,{\widehat{b}}_p)$ and $({\widehat{a}}_q,b_q)$. The two projections $E\to A$ and $E\to B$ are surjective, since their images contain the chosen algebra generators. Every element $(x,y)\in A{\times }_{C}B$ has a common image $P(c_1,\dots ,c_r)$ for a polynomial $P$ over $k$. Subtracting $P(({\alpha }_1,{\beta }_1),\dots ,({\alpha }_r,{\beta }_r))\in E$ leaves $(i,j)$ with $i\in I$ and $j\in J$. Write $i={\sum }_{\mu }{d}_{\mu }{i}_{\mu }$ and $j={\sum }_{\nu }{e}_{\nu }{j}_{\nu }$, with $d_{\mu }\in A$ and $e_{\nu }\in B$. By the surjectivity of the projections, choose $(d_{\mu },d_{\mu }^{\mathrm{\prime }})\in E$ and $(e_{\nu }^{\mathrm{\prime }},e_{\nu })\in E$. Then

Thus every $(x,y)$ belongs to $E$, so $E=A{\times }_{C}B$. Hence $A{\times }_{C}B$ is a finitely generated $k$-algebra, and $\mathrm{Spec}(A{\times }_{C}B)$ is an affine open neighbourhood of $z$ whose coordinate ring is of finite type over $k$.

Let $t\in \mathrm{\mid }T\mathrm{\mid }\setminus \mathrm{\mid }Z\mathrm{\mid }$. By [F4] part (3) the point $t$ lies in $X_1\setminus Z$ or in $X_2\setminus Z$; say $t\in X_1\setminus Z$, an open subscheme of $X_1$ and of $T$, on which the structure sheaf of $T$ restricts to that of $X_1$ and the structure morphism to $\mathrm{Spec}k$ restricts to the one of $X_1$. By [F16] choose an affine open neighbourhood $W=\mathrm{Spec}R\subseteq X_1$ of $t$. The intersection $W\cap (X_1\setminus Z)$ is an open neighbourhood of $t$ in $W$, so by the basis statement [F22] there is $f\in R$ with $t\in D(f)\subseteq W\cap (X_1\setminus Z)$. By [F23] the open subscheme $D(f)$ is affine with coordinate ring $R_f$, which is a finitely generated $k$-algebra: $R$ is finitely generated over $k$ by the affine-locality [F12] applied to the locally finite type morphism $X_1\to \mathrm{Spec}k$ of step 1.3, and this principal localization of a finitely generated $k$-algebra is finitely generated, being a quotient of a polynomial ring in one further variable by [F32]. Thus every point of $T\setminus Z$ has an affine open neighbourhood with finitely generated coordinate ring over $k$.

Let $h:\mathrm{Spec}R\to T$ be a morphism of the kind considered in step 1.5 and let $X\subseteq T$ be a closed subscheme with $h(\eta )\in \mathrm{\mid }X\mathrm{\mid }$. Then $h$ factors through $X$. Indeed $h^{-1}(\mathrm{\mid }X\mathrm{\mid })$ is a closed subset of $\mathrm{Spec}R$ containing $\eta$, so it contains $\overline{\{\eta \}}=\mathrm{Spec}R$ by step 1.5; hence $h(\mathrm{Spec}R)\subseteq \mathrm{\mid }X\mathrm{\mid }$. Choose an affine open $W=\mathrm{Spec}A\subseteq T$ containing the image of the closed point of $\mathrm{Spec}R$, which exists by [F16], and write $X\cap W=\mathrm{Spec}(A\mathrm{/}J)$ with the ideal $J$ provided by [F6]. By step 1.5 the preimage $h^{-1}(W)$ is all of $\mathrm{Spec}R$; so $h$ restricts to a morphism $\mathrm{Spec}R\to W$, corresponding under [F17] to a ring map $\psi :A\to R$, the global sections of $\mathrm{Spec}R$ being $R$ by [F19]. The image of the generic point is ${\psi }^{-1}(0)=\ker \psi$ by [F18], and it lies in $\mathrm{\mid }X\mathrm{\mid }\cap W=V(J)$ by [F20], so $J\subseteq \ker \psi$ and $\psi$ factors as $A\to A\mathrm{/}J\to R$; the corresponding morphism $\mathrm{Spec}R\to X\cap W$ has composite with the restrictions $X\cap W\to W\to T$ equal to $h$ by the bijection [F17], since both sides induce the same ring map $A\to R$.

The charts of steps 2.2 and 2.3 cover $T$: every point of $T$ lies in $\mathrm{\mid }Z\mathrm{\mid }$, so in one of the charts $\mathrm{Spec}(A{\times }_{C}B)$, or outside $\mathrm{\mid }Z\mathrm{\mid }$, so in one of the affine opens $D(f)\subseteq X_1\setminus Z$, $X_2\setminus Z$. All of these affine opens lie over the single affine open $\mathrm{Spec}k$ of the target, and their coordinate rings are finitely generated $k$-algebras; by the affine-locality [F12] the morphism $T\to \mathrm{Spec}k$ is locally of finite type.

Let $U,V\subseteq T$ be affine opens; they lie over the common affine open $\mathrm{Spec}k$ of the base. By step 1.4 the subschemes $U\cap X_1$ and $V\cap X_1$ are affine opens of $X_1$, and $X_1\to \mathrm{Spec}k$ is separated by step 1.3, so [F13] gives that $U\cap V\cap X_1=(U\cap X_1)\cap (V\cap X_1)$ is affine, hence quasi-compact by [F29]. The same argument shows that $U\cap V\cap X_2$ is quasi-compact. Since $\mathrm{\mid }T\mathrm{\mid }=\mathrm{\mid }{X}_1\mathrm{\mid }\cup \mathrm{\mid }{X}_2\mathrm{\mid }$ by step 1.2, the open set $U\cap V$ is the union of the two quasi-compact subspaces $U\cap V\cap X_1$ and $U\cap V\cap X_2$, hence quasi-compact by the finite-union argument of step 2.1. Therefore $T\to \mathrm{Spec}k$ is quasi-separated in the sense of [F10].

Let a valuative diagram for $T\to \mathrm{Spec}k$ be given: a valuation ring $R$ with fraction field $K$, a morphism $\mathrm{Spec}K\to T$ and a morphism $\mathrm{Spec}R\to \mathrm{Spec}k$ forming a commutative square [F15]. Let $p\in \mathrm{\mid }T\mathrm{\mid }$ be the image of the generic point. Since $\mathrm{\mid }T\mathrm{\mid }=\mathrm{\mid }{X}_1\mathrm{\mid }\cup \mathrm{\mid }{X}_2\mathrm{\mid }$ by step 1.2, after possibly interchanging the indices $1,2$ we may assume $p\in \mathrm{\mid }{X}_1\mathrm{\mid }$. Choosing an affine open $W=\mathrm{Spec}A\subseteq T$ around $p$ with $X_1\cap W=\mathrm{Spec}(A\mathrm{/}J)$ by [F6, F16], the generic morphism $\mathrm{Spec}K\to T$ corresponds to a ring map $A\to K$ whose kernel is $p$ [F17, F18] and which therefore kills $J$; as in step 2.4 it follows that the generic morphism factors as $\mathrm{Spec}K\to X_1\to T$ through the closed immersion $a$. Consequently the generic morphism $\mathrm{Spec}K\to X_1$, together with the given $\mathrm{Spec}R\to \mathrm{Spec}k$, is a valuative diagram for the proper morphism $X_1\to \mathrm{Spec}k$ of step 1.3: the square commutes because $a$ is a morphism of $k$-schemes, so the two composites $\mathrm{Spec}K\to \mathrm{Spec}k$ agree. The morphism $X_1\to \mathrm{Spec}k$ is of finite type by step 1.3, and is quasi-separated because separatedness makes its affine-open intersections affine by [F13], hence quasi-compact by [F29]. Thus [F14] supplies a lift $\mathrm{Spec}R\to X_1$ of that diagram; composing it with $a$ gives a lift $\mathrm{Spec}R\to T$ of the original diagram, whose generic restriction is the given morphism because the lift in $X_1$ has the prescribed generic restriction.

Let $h_1,h_2:\mathrm{Spec}R\to T$ be two lifts of one valuative diagram for $T\to \mathrm{Spec}k$ [F15]; we show $h_1=h_2$. By the definition of a lift both restrict to the given generic morphism, so with $\eta$ the generic point of $\mathrm{Spec}R$ given by step 1.5 the points $h_1(\eta )=h_2(\eta )=p$ coincide, and by step 1.2 we may assume $p\in \mathrm{\mid }{X}_1\mathrm{\mid }$. Applying step 2.4 to $h_1$ and $h_2$ with the closed subscheme $X_1\subseteq T$ gives factorizations $h_i=a{h}_i^{\mathrm{\prime }}$ with $h_i^{\mathrm{\prime }}:\mathrm{Spec}R\to X_1$. The closed immersion $a$ is a monomorphism by [F28], so from $a(h_1^{\mathrm{\prime }}){\mathrm{\mid }}_{\mathrm{Spec}K}=h_1{\mathrm{\mid }}_{\mathrm{Spec}K}=h_2{\mathrm{\mid }}_{\mathrm{Spec}K}=a(h_2^{\mathrm{\prime }}){\mathrm{\mid }}_{\mathrm{Spec}K}$ we get $(h_1^{\mathrm{\prime }}){\mathrm{\mid }}_{\mathrm{Spec}K}=(h_2^{\mathrm{\prime }}){\mathrm{\mid }}_{\mathrm{Spec}K}$. Hence $h_1^{\mathrm{\prime }}$ and $h_2^{\mathrm{\prime }}$ are two lifts of one valuative diagram for the proper morphism $X_1\to \mathrm{Spec}k$ of step 1.3, which is of finite type and quasi-separated; by the uniqueness assertion of [F14] (which holds for every valuative diagram of a proper morphism) $h_1^{\mathrm{\prime }}=h_2^{\mathrm{\prime }}$, and therefore $h_1=h_2$.

By step 2.1 the morphism $T\to \mathrm{Spec}k$ is quasi-compact and by step 3.1 it is locally of finite type; by [F9] and [F12] it is of finite type.

Steps 3.3 and 3.4 show that every valuative diagram for $T\to \mathrm{Spec}k$ over an arbitrary valuation ring has exactly one lift. By step 4.1 the morphism $T\to \mathrm{Spec}k$ is of finite type and by step 3.2 it is quasi-separated, so the converse direction of the criterion [F14] shows that $T\to \mathrm{Spec}k$ is proper. Combined with the existence clause of step 1.2 this proves that the closed-subscheme pushout $X_1{⨿}_Z{X}_2$ exists as a $k$-scheme, that each $X_i$ is a closed subscheme of it, and that it is proper over $k$, as claimed.

The Axiom of Choice [F1] is used exactly through the four cited results [F3], [F4], [F6] and [F14], each of which assumes it; every other step selects only finitely many objects (finitely many generators, finitely many members of a finite subcover) and is choice-free. The statement has no degenerate case: $k$ is nonempty, the line and the conic are nonempty subschemes of $X_i$, so $Z$ is nonempty, and the argument above nowhere uses properness or nonemptiness of $Z$ beyond the closed-immersion hypotheses.