Pushouts of closed immersions exist

Let $\mathrm{\mid }T\mathrm{\mid }$ be the quotient of the disjoint union $\mathrm{\mid }X\mathrm{\mid }\bigsqcup \mathrm{\mid }Y\mathrm{\mid }$ by the equivalence relation generated by $i(z)\sim j(z)$ for $z\in \mathrm{\mid }Z\mathrm{\mid }$, equipped with the quotient topology of [F8]. Since $i$ and $j$ are injective homeomorphisms onto closed subsets by [F1], distinct points of $\mathrm{\mid }X\mathrm{\mid }\setminus \mathrm{\mid }Z\mathrm{\mid }$ and $\mathrm{\mid }Y\mathrm{\mid }\setminus \mathrm{\mid }Z\mathrm{\mid }$ are inequivalent and each of $\mathrm{\mid }X\mathrm{\mid }$, $\mathrm{\mid }Y\mathrm{\mid }$ maps homeomorphically onto a closed subset of $\mathrm{\mid }T\mathrm{\mid }$, with $\mathrm{\mid }T\mathrm{\mid }=\mathrm{\mid }X\mathrm{\mid }\cup \mathrm{\mid }Y\mathrm{\mid }$ and $\mathrm{\mid }X\mathrm{\mid }\cap \mathrm{\mid }Y\mathrm{\mid }=\mathrm{\mid }Z\mathrm{\mid }$; moreover a map $\mathrm{\mid }T\mathrm{\mid }\to W$ is continuous if and only if its two restrictions to $\mathrm{\mid }X\mathrm{\mid }$ and $\mathrm{\mid }Y\mathrm{\mid }$ are continuous, because $\mathrm{\mid }T\mathrm{\mid }$ carries the quotient topology.

For open $U\subseteq \mathrm{\mid }T\mathrm{\mid }$ set ${\mathcal{O}}_T(U)=\{(s,t)\in {\mathcal{O}}_X(U\cap X)\times {\mathcal{O}}_Y(U\cap Y):s{\mathrm{\mid }}_{U\cap Z}=t{\mathrm{\mid }}_{U\cap Z}\}$, with componentwise restriction maps. This presheaf is a sheaf of commutative rings in the sense of [F7]: locality is componentwise, and for compatible local sections $(s_{\alpha },t_{\alpha })$ the sections $s_{\alpha }$ and $t_{\alpha }$ glue to sections $s$ and $t$ of ${\mathcal{O}}_X$ and ${\mathcal{O}}_Y$ whose restrictions to the open cover of $U\cap Z$ agree, so that $s{\mathrm{\mid }}_{U\cap Z}=t{\mathrm{\mid }}_{U\cap Z}$. Hence $(\mathrm{\mid }T\mathrm{\mid },{\mathcal{O}}_T)$ is a ringed space, and by [F9] and [F6] its structure sheaf is the fibre product $a_{\ast }{\mathcal{O}}_X{\times }_{c_{\ast }{\mathcal{O}}_Z}{b}_{\ast }{\mathcal{O}}_Y$ of direct images, where $a,b,c$ are induced by the inclusions of $\mathrm{\mid }X\mathrm{\mid },\mathrm{\mid }Y\mathrm{\mid },\mathrm{\mid }Z\mathrm{\mid }$ in $\mathrm{\mid }T\mathrm{\mid }$.

We compute the stalks in the sense of [F10]. If $t\in \mathrm{\mid }T\mathrm{\mid }$ is not in $\mathrm{\mid }Z\mathrm{\mid }$, then $t$ lies in exactly one of the open subsets $\mathrm{\mid }X\mathrm{\mid }\setminus \mathrm{\mid }Z\mathrm{\mid }$, $\mathrm{\mid }Y\mathrm{\mid }\setminus \mathrm{\mid }Z\mathrm{\mid }$, on which ${\mathcal{O}}_T$ restricts to ${\mathcal{O}}_X$ resp. ${\mathcal{O}}_Y$, so ${\mathcal{O}}_{T,t}={\mathcal{O}}_{X,t}$ or ${\mathcal{O}}_{T,t}={\mathcal{O}}_{Y,t}$. If $t\in \mathrm{\mid }Z\mathrm{\mid }$, a germ of ${\mathcal{O}}_T$ at $t$ is a pair of germs in ${\mathcal{O}}_{X,t}\times {\mathcal{O}}_{Y,t}$ whose images in ${\mathcal{O}}_{Z,t}$ coincide, and the filtered colimit of the fibre products of the ${\mathcal{O}}_X(U\cap X)$ and ${\mathcal{O}}_Y(U\cap Y)$ over ${\mathcal{O}}_Z(U\cap Z)$ is the fibre product of the colimits, so ${\mathcal{O}}_{T,t}={\mathcal{O}}_{X,t}{\times }_{{\mathcal{O}}_{Z,t}}{\mathcal{O}}_{Y,t}$. The maps ${\mathcal{O}}_{X,t}\to {\mathcal{O}}_{Z,t}$ and ${\mathcal{O}}_{Y,t}\to {\mathcal{O}}_{Z,t}$ are surjective because $i,j$ are closed immersions by [F1]; their kernels are proper ideals since the target stalk is a nonzero local ring, hence lie in the respective maximal ideals by [F5].

For every $z\in \mathrm{\mid }Z\mathrm{\mid }$ there exist affine opens $U=\mathrm{Spec}A\subseteq X$ and $V=\mathrm{Spec}B\subseteq Y$ with $i(z)\in U$, $j(z)\in V$ and $i^{-1}(U)=j^{-1}(V)$. Choose affine opens $U_0=\mathrm{Spec}{A}_0$ of $X$ containing $i(z)$ and $V_0=\mathrm{Spec}{B}_0$ of $Y$ containing $j(z)$, which exist by [F3]. By [F2] the preimages $i^{-1}(U_0)=\mathrm{Spec}(A_0\mathrm{/}{J}_0)$ and $j^{-1}(V_0)=\mathrm{Spec}(B_0\mathrm{/}{J}_0^{\mathrm{\prime }})$ are affine open subschemes of $Z$ containing $z$, and $z$ lies in the open subset $i^{-1}(U_0)\cap j^{-1}(V_0)$ of $Z$. Since distinguished subsets form a basis of the topology of an affine spectrum by [F12], choose $h\in A_0\mathrm{/}{J}_0$ with $z\in D(h)\subseteq i^{-1}(U_0)\cap j^{-1}(V_0)$, and lift $h$ to some $u\in A_0$, possible because $A_0\to A_0\mathrm{/}{J}_0$ is surjective by [F2]. Then $U=D(u)\subseteq X$ is an affine open with $i(z)\in U$ and $i^{-1}(U)=D(h)$, because inside the closed subscheme $\mathrm{Spec}(A_0\mathrm{/}{J}_0)$ of the affine scheme $U_0$ the trace of $D(u)$ is $D(u<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>J_0)=D(h)$; in particular $i^{-1}(U)\subseteq j^{-1}(V_0)$. Again by [F12], applied to the affine scheme $j^{-1}(V_0)=\mathrm{Spec}(B_0\mathrm{/}{J}_0^{\mathrm{\prime }})$, choose $g\in B_0\mathrm{/}{J}_0^{\mathrm{\prime }}$ with $z\in D(g)\subseteq i^{-1}(U)$, and lift $g$ to some $v\in B_0$ by [F2]; then $V=D(v)\subseteq Y$ is affine with $j(z)\in V$ and $j^{-1}(V)=D(g)\subseteq i^{-1}(U)$, the trace of $D(v)$ on $\mathrm{Spec}(B_0\mathrm{/}{J}_0^{\mathrm{\prime }})$ being $D(g)$. Finally $i^{-1}(U)$ is an open subscheme of $j^{-1}(V_0)$ and of $U$, so $i^{-1}(U)\hookrightarrow j^{-1}(V_0)\to V_0$ is a morphism of affines; by [F2] applied to the closed immersion $i:Z\to X$ and the affine open $U$ we have $i^{-1}(U)=\mathrm{Spec}(A\mathrm{/}I)$ with $A=\mathrm{\Gamma }(U,\mathcal{O})=A_u$ and $I=\ker (A\to C)$, $C=\mathrm{\Gamma }(i^{-1}(U),\mathcal{O})$, the restriction $A\to C$ being surjective. The section $g$ restricts to an element $\bar{g}\in C=A\mathrm{/}I$, and $\bar{g}$ lifts to some $f\in A$ by surjectivity. The trace of $D(f)$ on the closed subscheme $i^{-1}(U)=\mathrm{Spec}(A\mathrm{/}I)$ of $U$ is $D(f<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>I)=D(\bar{g})$, and $D(\bar{g})=D(g)$ as open subsets of $i^{-1}(U)$, since $\bar{g}$ is the restriction of the function $g$ and principal opens restrict to traces of principal opens. Hence $U^{\mathrm{\prime }}=D(f)\subseteq X$ and $V=D(v)\subseteq Y$ are affine opens with $i(z)\in U^{\mathrm{\prime }}$, $j(z)\in V$ and $i^{-1}(U^{\mathrm{\prime }})=D(\bar{g})=D(g)=j^{-1}(V)$, as required.

The ring $R_t={\mathcal{O}}_{X,t}{\times }_{{\mathcal{O}}_{Z,t}}{\mathcal{O}}_{Y,t}$ of step 1.3 is local with maximal ideal ${\mathfrak{m}}_{X,t}{\times }_{{\mathcal{O}}_{Z,t}}{\mathfrak{m}}_{Y,t}=:\mathfrak{m}$ and residue field the common residue field $k(t)$: an element $(s,u)$ with $s\mathrm{\notin }{\mathfrak{m}}_{X,t}$ is a unit, because then the image of $s$ in ${\mathcal{O}}_{Z,t}$ is a unit and $u$ has the same image as $s$ modulo the kernel of ${\mathcal{O}}_{Y,t}\to {\mathcal{O}}_{Z,t}$, which lies in ${\mathfrak{m}}_{Y,t}$, so that $u$ is a unit as well; the ideal $\mathfrak{m}$ is proper because $(1,1)\mathrm{\notin }\mathfrak{m}$, the ring $R_t$ is nonzero because $1=(1,1)\mathrm{\ne }0$, and every element outside $\mathfrak{m}$ has a component outside ${\mathfrak{m}}_{X,t}$ or ${\mathfrak{m}}_{Y,t}$ and is therefore invertible, so that $\mathfrak{m}$ is the unique maximal ideal of the local ring $R_t$ by [F5]. Hence every stalk of ${\mathcal{O}}_T$ is a local ring, $T$ is a locally ringed space by [F4], and $a:X\to T$, $b:Y\to T$, $c:Z\to T$ are morphisms of locally ringed spaces by [F11].

In the situation of step 1.4 write $C$ for the ring of the common closed subscheme $U\cap Z=V\cap Z$, so that there are surjections $A\to C$ and $B\to C$ by [F2], and put $R=A{\times }_{C}B$, the fibre product of rings with coordinatewise operations. Let $W\subseteq T$ be the image of $\mathrm{\mid }U\mathrm{\mid }\bigsqcup \mathrm{\mid }V\mathrm{\mid }$. Every prime of $R$ contains $\ker (R\to A)=0\times \ker (B\to C)$ or $\ker (R\to B)=\ker (A\to C)\times 0$, since the product of an element of the first kernel with an element of the second kernel is zero; hence every prime of $R$ is the preimage of a prime of $A$ or of $B$; the two closed images intersect exactly in $\mathrm{Spec}C$. Since these closed images cover $\mathrm{Spec}R$, they give the quotient topology, so $\mathrm{\mid }\mathrm{Spec}R\mathrm{\mid }=\mathrm{\mid }U\mathrm{\mid }{\bigsqcup }_{\mathrm{\mid }U\cap Z\mathrm{\mid }}\mathrm{\mid }V\mathrm{\mid }$ as a topological space, which is the topology of $W$ from step 1.1. Moreover $R_{(a,b)}\cong A_a{\times }_{C_{\bar{a}}}{B}_b$ for $(a,b)\in R$, with the restrictions of $a$ and $b$ agreeing in $C$; under this identification the sections of ${\mathcal{O}}_T$ over the basic open $D((a,b))$ of $W$ are exactly $\mathrm{\Gamma }(D(a),{\mathcal{O}}_U){\times }_{\mathrm{\Gamma }(D(\bar{a}),{\mathcal{O}}_Z)}\mathrm{\Gamma }(D(b),{\mathcal{O}}_V)$, which by [F13] equals $A_a{\times }_{C_{\bar{a}}}{B}_b$. As the distinguished opens form a basis, the identity on $\mathrm{\mid }\mathrm{Spec}R\mathrm{\mid }=\mathrm{\mid }W\mathrm{\mid }$ extends to an isomorphism of ringed spaces $W\cong \mathrm{Spec}R$, so each point of $\mathrm{\mid }Z\mathrm{\mid }$ has an affine open neighbourhood in $T$.

Points of $T$ outside $\mathrm{\mid }Z\mathrm{\mid }$ lie in the open subschemes $X\setminus Z$ and $Y\setminus Z$, which are covered by affine opens of $X$ and of $Y$ disjoint from $Z$; together with step 2.2 this shows that every point of $T$ has an affine open neighbourhood, so $T$ is a scheme by [F3]; moreover $T$ is a scheme over $S$: writing $f:X\to S$ and $g:Y\to S$ for the given structure morphisms, $fi=gj$, so step 1.1 gives a continuous map $h:\mathrm{\mid }T\mathrm{\mid }\to \mathrm{\mid }S\mathrm{\mid }$ with $ha=f$ and $hb=g$, the sheaf maps ${\mathcal{O}}_S\to (ha{)}_{\ast }{\mathcal{O}}_X$ and ${\mathcal{O}}_S\to (hb{)}_{\ast }{\mathcal{O}}_Y$ induced by $f$ and $g$ agree on $(hc{)}_{\ast }{\mathcal{O}}_Z$ because $fi=gj$ and hence define a map ${\mathcal{O}}_S\to h_{\ast }{\mathcal{O}}_T$ by the fibre product description of step 1.2, which is local at every point by step 2.1; thus $h$ is a morphism of locally ringed spaces over which $a$ and $b$ are morphisms.

The map $a:X\to T$ is a closed immersion. On an affine chart $W=\mathrm{Spec}(A{\times }_{C}B)$ of step 2.2 the preimage of $W$ in $X$ is $U=\mathrm{Spec}A$, and the induced ring map $A{\times }_{C}B\to A$ is the first projection, which is surjective: for $a\in A$ choose $b\in B$ lifting $\bar{a}\in C$, which is possible since $B\to C$ is surjective, and then $(a,b)\in A{\times }_{C}B$ maps to $a$. A surjective ring map induces a closed immersion of affine schemes by [F2]; the charts of step 2.2 together with the open subschemes $X\setminus Z$ and $Y\setminus Z$ cover $T$, so [F14] gives that $a$ is a closed immersion; the same argument applies to $b$. Since $\mathrm{\mid }X\mathrm{\mid }\cap \mathrm{\mid }Y\mathrm{\mid }=\mathrm{\mid }Z\mathrm{\mid }$ by step 1.1 and $A{\otimes }_{R}B\cong C$ for $R=A{\times }_{C}B$ with both maps to $C$ surjective, the square is Cartesian in each chart; fibre products of schemes exist by [F17], the Cartesian property is local on $T$, and the charts of step 2.2 together with the open subschemes $X\setminus Z$ and $Y\setminus Z$ cover $T$ by step 1.1, so the universal property of [F15] gives $Z\cong X{\times }_{T}Y$ over $T$.

$T$ is the pushout of $X$ and $Y$ over $Z$ in locally ringed spaces, hence in schemes and in $S$-schemes. Indeed, let $f:X\to W$ and $g:Y\to W$ be $S$-morphisms with $fi=gj$. Their underlying maps agree on $\mathrm{\mid }Z\mathrm{\mid }$ and hence induce a unique continuous map $h:\mathrm{\mid }T\mathrm{\mid }\to \mathrm{\mid }W\mathrm{\mid }$ by the quotient property of step 1.1; the maps $f^{\mathrm{♯}}:{\mathcal{O}}_W\to (ha{)}_{\ast }{\mathcal{O}}_X$ and $g^{\mathrm{♯}}:{\mathcal{O}}_W\to (hb{)}_{\ast }{\mathcal{O}}_Y$ agree on the direct image of ${\mathcal{O}}_Z$ because $fi=gj$, hence define a map of sheaves ${\mathcal{O}}_W\to h_{\ast }{\mathcal{O}}_T=h_{\ast }{a}_{\ast }{\mathcal{O}}_X{\times }_{h_{\ast }{c}_{\ast }{\mathcal{O}}_Z}{h}_{\ast }{b}_{\ast }{\mathcal{O}}_Y$ by step 1.2, and this map is local at every point by step 2.1, using that the stalk maps of $f$ and $g$ are local. Thus $(h,h^{\mathrm{♯}})$ is a morphism of locally ringed spaces, and it is the unique one compatible with $f$ and $g$ because $\mathrm{\mid }T\mathrm{\mid }$ carries the quotient topology and ${\mathcal{O}}_T$ injects into the product of the two direct images by step 1.2; if $W$ and the morphisms $f,g$ are over $S$, then the composite $T\to W\to S$ agrees with the structure morphism $T\to S$ of step 3.1, because both agree after composing with $a$ and with $b$ and $\mathrm{\mid }T\mathrm{\mid }=\mathrm{\mid }X\mathrm{\mid }\cup \mathrm{\mid }Y\mathrm{\mid }$, so the morphism is one of $S$-schemes.

The Axiom of Choice [F16] is assumed in the statement and enters exactly through [F2], which assumes AC: it is used in step 1.4 for the affine form of a closed immersion and in steps 2.2 and 3.2 for the chart computations. All other arguments are choice-free: the charts and germs are exhibited from the given data point by point, without selecting from any family, and no other cited item uses a choice principle. The degenerate cases are included: if $Z=\mathrm{\varnothing }$ then $T$ is the disjoint union $X\bigsqcup Y$, which is the empty gluing case of the construction; if one of the closed immersions is an isomorphism, the pushout is the other scheme, which the chart computations reproduce; and the constructions are empty-ready, since the empty affine scheme corresponds to the zero ring by [F2]. ∎