7-1 閉埋入・開埋入・局所閉埋入

localizationのprime correspondenceにより $\operatorname{Spec}A_f$ のpointsはprimes $\mathfrak p\subset A$ with $f\notin\mathfrak p$、すなわち $D(f)$ のpointsとbijectionします。principal opens $D_{A_f}(a/f^n)$ は $D_A(a)\cap D_A(f)$ に対応するためhomeomorphismです。
structure sheavesについて、corresponding point $\mathfrak p$ at stalk levelでは
$$ (A_f)_{\mathfrak pA_f}\cong A_{\mathfrak p}. $$
mapはlocal fractionsを同じfractionへ送ります。sheaf morphismがall stalksでisomorphismならsheaf isomorphismなので、$\operatorname{Spec}A_f$ is isomorphic to the restricted scheme on $D(f)$。□

$D(f)\cap V(I)$ corresponds to the principal open $D(\bar f)$ of $\operatorname{Spec}(A/I)$。sections are
$$ (A/I)_{\bar f}. $$
localization is exact for modules, and directly the map
$$ A_f\longrightarrow(A/I)_{\bar f}, \qquad \frac a{f^n}\longmapsto\frac{\bar a}{\bar f^n} $$
is surjective with kernel $I_f$。従ってclaim holds on principal-open basis. Surjectivity and kernel equality can be checked on stalks or a basis, so they hold as sheaf statements. □

On each affine open $V=\operatorname{Spec}A$, the inverse image $i^{-1}(V)=\operatorname{Spec}(A/I_V)$ maps homeomorphically to $V(I_V)$, and the quotient-map proposition gives sheaf surjectivity over $V$。These local closed subsets agree on overlaps because they are all the image of $X$。
The complement of the image has intersection $V\setminus V(I_V)$ with every member $V$ of an affine open cover of $Y$。Each such intersection is open, so the complement is open and the global image is closed. The local homeomorphisms to $V(I_V)$ agree with the same underlying map $i$, hence glue to a homeomorphism onto the image. Finally, sheaf surjectivity is detected on stalks; every stalk lies in some affine $V$, where surjectivity was proved. Thus both claims hold globally. □

affine open $V=\operatorname{Spec}A$ 上では、closed subscheme is $\operatorname{Spec}(A/I)$ for a unique ideal $I$。uniqueness follows because the kernel of $A\to A/I$ is $I$。On overlap principal opens, ideals localize: $I|_{D(f)}=I_f$。Therefore local ideals glue to a quasi-coherent ideal sheaf.
Conversely given $\mathcal I$, on each affine $V=\operatorname{Spec}A$ choose $I$ with $\mathcal I|_V=\widetilde I$ and form $\operatorname{Spec}(A/I)$。On overlaps these schemes agree because after refining by principal opens both are quotients by the same localized ideal. The gluing theorem produces a closed subscheme. Applying either construction twice recovers each local ideal and quotient, hence recovers the global object uniquely. □

radical ideals localize compatibly:
$$ S^{-1}\sqrt I=\sqrt{S^{-1}I}. $$
Indeed $a/s$ has a power in $S^{-1}I$ iff some $t\in S$ satisfies $ta^n\in I$, equivalently $ta\in\sqrt I$ after possibly replacing $t$ by a power. Thus the local quotient schemes glue. Each $A/\sqrt I$ is reduced and has zero locus $V(I)$。
If another reduced closed subscheme on $Z$ is given locally by $A/J$, then $V(J)=V(I)$ implies $\sqrt J=\sqrt I$。Reducedness makes $J=\sqrt J$。Hence $J=\sqrt I$, proving uniqueness. □

fiber product ring is
$$ (A/I)\otimes_A(A/J)\cong A/(I+J), $$
where $\bar a\otimes\bar b\mapsto\overline{ab}$ and inverse $\bar c\mapsto\bar c\otimes1$。Thus intersection formula follows.
For union,
$$ V(I\cap J)=V(IJ)=V(I)\cup V(J) $$
as point sets. The ideal $I\cap J$ is the largest ideal contained in both $I$ and $J$。Therefore the quotient $A/(I\cap J)$ maps to both $A/I$ and $A/J$ and gives the smallest closed subscheme containing both. This universal property defines scheme-theoretic union. □

factor $i=j\circ c$ as in the definition. The image of $c$ is closed in $U$, hence $U\cap F$ for its closure $F$ in $Y$, so it is locally closed in $Y$。The homeomorphism and sheaf surjection follow from the closed immersion criterion inside $U$。
Conversely, if $Z$ is closed in an open $U\subseteq Y$ and is equipped with a closed subscheme structure $X\hookrightarrow U$, composing with $U\hookrightarrow Y$ gives the required factorization. □

For open immersion $j:U\hookrightarrow Y$, a morphism $T\to Y$ whose image lies in $U$ factors uniquely through restricted locally ringed space $U$。Thus two factorizations are equal.
For closed immersion, work locally with $X=\operatorname{Spec}(A/I)$ and $Y=\operatorname{Spec}A$。Two maps $T\to X$ with same composite to $Y$ induce, on every affine open of $T$, the same ring map from $A$; because it kills $I$, the factorization through $A/I$ is unique. Therefore the maps agree locally and hence globally.
A locally closed immersion is a composite of a closed and an open immersion. A composite of monomorphisms is a monomorphism, proving all cases. □

For open immersions, composition corresponds to inclusion of open subsets. Under base change $Y'\to Y$, the pullback of $U\subseteq Y$ has underlying open subset inverse image of $U$ in $Y'$ and restricted structure sheaf, hence is an open immersion.
For closed immersions, composition is affine-locally composition of surjections
$$ A\twoheadrightarrow A/I\twoheadrightarrow(A/I)/J, $$
again a quotient of $A$。For base change by $A\to B$,
$$ (A/I)\otimes_AB\cong B/IB, $$
so the pullback is the closed immersion $\operatorname{Spec}(B/IB)\hookrightarrow\operatorname{Spec}B$。These affine computations glue.
For locally closed immersions, write $X\hookrightarrow U\hookrightarrow Y$ as closed then open. Base change each factor to obtain closed then open again. For composition, suppose $X$ is closed in $U$ with $U$ open in $Y$, and $Y$ is closed in $V$ with $V$ open in $Z$。Because $U$ is open in the subspace $Y$, there is an open $W\subseteq V$ with $U=Y\cap W$。Then $U$ is closed in $W$, and $X$ is closed in $U$, hence $X$ is closed in $W$。Since $W$ is open in $Z$, this factors $X\to Z$ as a closed immersion followed by an open immersion. Thus the composite is locally closed. □