Schematic closure and agreement on a dense open

For an open $W\subseteq Y$ the map $\mathcal{K}(W)\to {\mathcal{O}}_Y(W)$ identifies $\mathcal{K}(W)$ with the set of sections $s\in {\mathcal{O}}_Y(W)$ satisfying $s{\mathrm{\mid }}_{U\cap W}=0$, because $(j_{\ast }{\mathcal{O}}_U)(W)={\mathcal{O}}_U(U\cap W)={\mathcal{O}}_Y(U\cap W)$ and the map is the restriction; hence $\mathcal{K}$ is an ideal sheaf on $Y$, and $\mathcal{K}{\mathrm{\mid }}_U=0$ because for $W\subseteq U$ the restriction ${\mathcal{O}}_Y(W)\to {\mathcal{O}}_Y(W\cap U)={\mathcal{O}}_Y(W)$ is the identity.

The scheme $Y$ has Noetherian underlying space by [F2], so every subspace of $Y$ is compact; for an affine open $V=\mathrm{Spec}A\subseteq Y$ the set $U\cap V$ is therefore compact, and since the distinguished opens contained in the open set $U\cap V$ form an open cover of it by [F3], compactness yields finitely many $f_1,\dots ,f_r\in A$ with $U\cap V=D(f_1)\cup \cdots \cup D(f_r)$, the empty family $r=0$ being allowed when $U\cap V=\mathrm{\varnothing }$.

By the sheaf axiom [F3] the restriction map $\mathrm{\Gamma }(U\cap V,\mathcal{O})\to {\prod }_{i=1}^r{A}_{f_i}$ is injective, being the first map of an equalizer diagram, and the composite $A\to \mathrm{\Gamma }(U\cap V,\mathcal{O})\to {\prod }_i{A}_{f_i}$ is the product of the localisation maps $A\to A_{f_i}$; therefore $\mathcal{K}(V)=\ker (A\to \mathrm{\Gamma }(U\cap V,\mathcal{O}))=\ker (A\to {\prod }_{i=1}^r{A}_{f_i})$, the empty product being the zero ring and $\mathrm{\Gamma }(\mathrm{\varnothing },\mathcal{O})=0$.

Let $g\in A$. Since $U\cap D(g)=(U\cap V)\cap D(g)$ is covered by the distinguished opens $D(g{f}_i)=D(g)\cap D(f_i)$ of the affine scheme $D(g)=\mathrm{Spec}{A}_g$, step 2.1 applied to the affine open $D(g)$ gives $\mathcal{K}(D(g))=\ker (A_g\to {\prod }_{i=1}^r{A}_{g{f}_i})$; because localisation is exact and $(A_{f_i}{)}_g\cong A_{g{f}_i}$ by [F4], localising the exact sequence $0\to \mathcal{K}(V)\to A\to {\prod }_i{A}_{f_i}$ at $g$ identifies this kernel with the localisation $\mathcal{K}(V{)}_g\subseteq A_g$. Hence $\mathcal{K}(D(g))=\mathcal{K}(V{)}_g$ for every $g\in A$.

The sheaf $\mathcal{K}{\mathrm{\mid }}_V$ equals the associated sheaf $\widetilde{\mathcal{K}(V)}$: both are subsheaves of ${\mathcal{O}}_V$, and by step 3.1 and [F5] they have the same sections on every distinguished open $D(g)\subseteq V$, namely $\mathcal{K}(V{)}_g$, with the restriction maps induced by those of ${\mathcal{O}}_V$; since the distinguished opens form a basis of $V$ and membership in a subsheaf of ${\mathcal{O}}_V$ is tested on a cover, an open section $s\in {\mathcal{O}}_V(W)$ lies in $\mathcal{K}(W)$ if and only if all its restrictions to distinguished opens inside $W$ lie in the corresponding $\mathcal{K}(V{)}_g$, which is exactly the condition that $s$ lies in $\widetilde{\mathcal{K}(V)}(W)$; hence $\mathcal{K}{\mathrm{\mid }}_V=\widetilde{\mathcal{K}(V)}$. As every point of $Y$ has such an affine neighbourhood $V$, the ideal sheaf $\mathcal{K}$ is quasi-coherent in the sense of [F6].

By step 4.1 and [F6] the sheaf $\mathcal{K}$ is a quasi-coherent ideal sheaf, so the correspondence [F7] applies: $Z:=Z_{\mathcal{K}}=(V(\mathcal{K}),({\mathcal{O}}_Y\mathrm{/}\mathcal{K}){\mathrm{\mid }}_{V(\mathcal{K})})$ is a closed subscheme $i:Z\to Y$ with $\ker ({\mathcal{O}}_Y\to i_{\ast }{\mathcal{O}}_Z)=\mathcal{K}$ and underlying closed set $V(\mathcal{K})=\{y:{\mathcal{K}}_y\mathrm{\ne }{\mathcal{O}}_{Y,y}\}$; moreover, on an affine open $V=\mathrm{Spec}A\subseteq Y$, the unique ideal of [F8] describing the closed immersion $i^{-1}(V)\to V$ is $\mathcal{K}(V)$, because it is the kernel of $A\to i_{\ast }{\mathcal{O}}_Z(V)={\mathcal{O}}_Z(Z\cap V)$ by [F1], so $Z\cap V$ is $\mathrm{Spec}(A\mathrm{/}\mathcal{K}(V))$ over $V$, with underlying set $V(\mathcal{K}(V))$.

For $y\in U$ step 1.1 gives ${\mathcal{K}}_y=0$, and ${\mathcal{O}}_{Y,y}$ is a nonzero local ring by [F10], so ${\mathcal{K}}_y\mathrm{\ne }{\mathcal{O}}_{Y,y}$ and $y\in V(\mathcal{K})=\mathrm{\mid }Z\mathrm{\mid }$; thus $U\subseteq \mathrm{\mid }Z\mathrm{\mid }$ as subsets of $Y$. Moreover ${\mathcal{O}}_Z{\mathrm{\mid }}_U=({\mathcal{O}}_Y\mathrm{/}\mathcal{K}){\mathrm{\mid }}_{V(\mathcal{K})}{\mathrm{\mid }}_U=({\mathcal{O}}_Y\mathrm{/}\mathcal{K}){\mathrm{\mid }}_U={\mathcal{O}}_U$, because restriction of a quotient is the quotient of the restrictions, $\mathcal{K}{\mathrm{\mid }}_U=0$ and ${\mathcal{O}}_Y{\mathrm{\mid }}_U={\mathcal{O}}_U$ by [F1]; consequently the open subspace of the scheme $Z$ on the open subset $U\subseteq \mathrm{\mid }Z\mathrm{\mid }$ is the scheme $U$ itself, and the inclusion $j^{\mathrm{\prime }}:U\to Z$ is an open immersion with $i\circ j^{\mathrm{\prime }}=j$ and $j^{\mathrm{\prime }}$ inducing the identity on structure sheaves over $U$.

To see that ${\mathcal{O}}_Z\to j_{\ast }^{\mathrm{\prime }}{\mathcal{O}}_U$ is injective, let $V=\mathrm{Spec}A\subseteq Y$ be affine; by step 5.1 the open $Z\cap V$ is $\mathrm{Spec}(A\mathrm{/}\mathcal{K}(V))$, so ${\mathcal{O}}_Z(Z\cap V)=A\mathrm{/}\mathcal{K}(V)$, the associated sheaf $(A\mathrm{/}\mathcal{K}(V){)}^{\sim }$ having support $V(\mathcal{K}(V))=Z\cap V$ and hence the same sections over $Z\cap V$ as over $V$; also $j_{\ast }^{\mathrm{\prime }}{\mathcal{O}}_U(Z\cap V)={\mathcal{O}}_U(U\cap (Z\cap V))={\mathcal{O}}_U(U\cap V)$ because $U\subseteq \mathrm{\mid }Z\mathrm{\mid }$ by step 6.1. Under these identifications the map is the ring map $A\mathrm{/}\mathcal{K}(V)\to \mathrm{\Gamma }(U\cap V,\mathcal{O})$ induced by the restriction $A\to \mathrm{\Gamma }(U\cap V,\mathcal{O})$, which is injective because that restriction has kernel $\mathcal{K}(V)$ by step 2.1. Since the affine opens $Z\cap V$ cover $Z$ and injectivity of a morphism of sheaves is local on a cover, ${\mathcal{O}}_Z\to j_{\ast }^{\mathrm{\prime }}{\mathcal{O}}_U$ is injective.

The subscheme $Z$ is the smallest closed subscheme of $Y$ through which $j$ factors: suppose $i^{\mathrm{\prime }}:Z^{\mathrm{\prime }}\to Y$ is a closed subscheme and $j=i^{\mathrm{\prime }}\circ s^{\mathrm{\prime }}$ for a morphism $s^{\mathrm{\prime }}:U\to Z^{\mathrm{\prime }}$; then the induced map ${\mathcal{O}}_Y\to j_{\ast }{\mathcal{O}}_U$ factors through the surjection ${\mathcal{O}}_Y\to i_{\ast }^{\mathrm{\prime }}{\mathcal{O}}_{Z^{\mathrm{\prime }}}$, so ${\mathcal{K}}^{\mathrm{\prime }}:=\ker ({\mathcal{O}}_Y\to i_{\ast }^{\mathrm{\prime }}{\mathcal{O}}_{Z^{\mathrm{\prime }}})\subseteq \mathcal{K}$, hence $V(\mathcal{K})\subseteq V({\mathcal{K}}^{\mathrm{\prime }})$; on an affine open $V=\mathrm{Spec}A$ the two closed subschemes are $Z\cap V=\mathrm{Spec}(A\mathrm{/}\mathcal{K}(V))$ and $Z^{\mathrm{\prime }}\cap V=\mathrm{Spec}(A\mathrm{/}{\mathcal{K}}^{\mathrm{\prime }}(V))$ with ${\mathcal{K}}^{\mathrm{\prime }}(V)\subseteq \mathcal{K}(V)$ by step 5.1, so the quotient map $A\mathrm{/}{\mathcal{K}}^{\mathrm{\prime }}(V)\to A\mathrm{/}\mathcal{K}(V)$ defines a closed immersion $Z\cap V\to Z^{\mathrm{\prime }}\cap V$ over $V$, and these maps agree on overlaps because both are the canonical quotient maps attached to the restricted ideals, so they glue to a morphism $Z\to Z^{\mathrm{\prime }}$ over $Y$; therefore every closed subscheme of $Y$ through which $j$ factors receives a morphism from $Z$, and, together with $j=i\circ j^{\mathrm{\prime }}$ from step 6.1, the closed subscheme $Z$ is the scheme-theoretic image of $j$ by [F11], that is, its schematic closure.

Finally let $T$ be a separated scheme and let $a,b:Z\to T$ be morphisms with $a\circ j^{\mathrm{\prime }}=b\circ j^{\mathrm{\prime }}$. By [F10] every scheme has a unique morphism to $\mathrm{Spec}\mathbb{Z}$, so $Z$ and $T$ are $\mathrm{Spec}\mathbb{Z}$-schemes, $a$ and $b$ are $\mathrm{Spec}\mathbb{Z}$-morphisms, and the structure morphism $T\to \mathrm{Spec}\mathbb{Z}$ is separated because the scheme $T$ is separated; since ${\mathcal{O}}_Z\to j_{\ast }^{\mathrm{\prime }}{\mathcal{O}}_U$ is injective by step 7.1 and $a,b$ agree on $U$, the agreement criterion [F9] gives $a=b$.