## 29.2 Closed immersions

In this section we elucidate some of the results obtained previously on closed immersions of schemes. Recall that a morphism of schemes $i : Z \to X$ is defined to be a closed immersion if (a) $i$ induces a homeomorphism onto a closed subset of $X$, (b) $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective, and (c) the kernel of $i^\sharp $ is locally generated by sections, see Schemes, Definitions 26.10.2 and 26.4.1. It turns out that, given that $Z$ and $X$ are schemes, there are many different ways of characterizing a closed immersion.

Lemma 29.2.1. Let $i : Z \to X$ be a morphism of schemes. The following are equivalent:

The morphism $i$ is a closed immersion.

For every affine open $\mathop{\mathrm{Spec}}(R) = U \subset X$, there exists an ideal $I \subset R$ such that $i^{-1}(U) = \mathop{\mathrm{Spec}}(R/I)$ as schemes over $U = \mathop{\mathrm{Spec}}(R)$.

There exists an affine open covering $X = \bigcup _{j \in J} U_ j$, $U_ j = \mathop{\mathrm{Spec}}(R_ j)$ and for every $j \in J$ there exists an ideal $I_ j \subset R_ j$ such that $i^{-1}(U_ j) = \mathop{\mathrm{Spec}}(R_ j/I_ j)$ as schemes over $U_ j = \mathop{\mathrm{Spec}}(R_ j)$.

The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$ and $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective.

The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective, and the kernel $\mathop{\mathrm{Ker}}(i^\sharp )\subset \mathcal{O}_ X$ is a quasi-coherent sheaf of ideals.

The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective, and the kernel $\mathop{\mathrm{Ker}}(i^\sharp )\subset \mathcal{O}_ X$ is a sheaf of ideals which is locally generated by sections.

**Proof.**
Condition (6) is our definition of a closed immersion, see Schemes, Definitions 26.4.1 and 26.10.2. So (6) $\Leftrightarrow $ (1). We have (1) $\Rightarrow $ (2) by Schemes, Lemma 26.10.1. Trivially (2) $\Rightarrow $ (3).

Assume (3). Each of the morphisms $\mathop{\mathrm{Spec}}(R_ j/I_ j) \to \mathop{\mathrm{Spec}}(R_ j)$ is a closed immersion, see Schemes, Example 26.8.1. Hence $i^{-1}(U_ j) \to U_ j$ is a homeomorphism onto its image and $i^\sharp |_{U_ j}$ is surjective. Hence $i$ is a homeomorphism onto its image and $i^\sharp $ is surjective since this may be checked locally. We conclude that (3) $\Rightarrow $ (4).

The implication (4) $\Rightarrow $ (1) is Schemes, Lemma 26.24.2. The implication (5) $\Rightarrow $ (6) is trivial. And the implication (6) $\Rightarrow $ (5) follows from Schemes, Lemma 26.10.1.
$\square$

Lemma 29.2.2. Let $X$ be a scheme. Let $i : Z \to X$ and $i' : Z' \to X$ be closed immersions and consider the ideal sheaves $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp )$ and $\mathcal{I}' = \mathop{\mathrm{Ker}}((i')^\sharp )$ of $\mathcal{O}_ X$.

The morphism $i : Z \to X$ factors as $Z \to Z' \to X$ for some $a : Z \to Z'$ if and only if $\mathcal{I}' \subset \mathcal{I}$. If this happens, then $a$ is a closed immersion.

We have $Z \cong Z'$ over $X$ if and only if $\mathcal{I} = \mathcal{I}'$.

**Proof.**
This follows from our discussion of closed subspaces in Schemes, Section 26.4 especially Schemes, Lemmas 26.4.5 and 26.4.6. It also follows in a straightforward way from characterization (3) in Lemma 29.2.1 above.
$\square$

Lemma 29.2.3. Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a sheaf of ideals. The following are equivalent:

$\mathcal{I}$ is locally generated by sections as a sheaf of $\mathcal{O}_ X$-modules,

$\mathcal{I}$ is quasi-coherent as a sheaf of $\mathcal{O}_ X$-modules, and

there exists a closed immersion $i : Z \to X$ of schemes whose corresponding sheaf of ideals $\mathop{\mathrm{Ker}}(i^\sharp )$ is equal to $\mathcal{I}$.

**Proof.**
The equivalence of (1) and (2) is immediate from Schemes, Lemma 26.10.1. If (1) holds, then there is a closed subspace $i : Z \to X$ with $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp )$ by Schemes, Definition 26.4.4 and Example 26.4.3. By Schemes, Lemma 26.10.1 this is a closed immersion of schemes and (3) holds. Conversely, if (3) holds, then (2) holds by Schemes, Lemma 26.10.1 (which applies because a closed immersion of schemes is a fortiori a closed immersion of locally ringed spaces).
$\square$

Lemma 29.2.4. The base change of a closed immersion is a closed immersion.

**Proof.**
See Schemes, Lemma 26.18.2.
$\square$

Lemma 29.2.5. A composition of closed immersions is a closed immersion.

**Proof.**
We have seen this in Schemes, Lemma 26.24.3, but here is another proof. Namely, it follows from the characterization (3) of closed immersions in Lemma 29.2.1. Since if $I \subset R$ is an ideal, and $\overline{J} \subset R/I$ is an ideal, then $\overline{J} = J/I$ for some ideal $J \subset R$ which contains $I$ and $(R/I)/\overline{J} = R/J$.
$\square$

Lemma 29.2.6. A closed immersion is quasi-compact.

**Proof.**
This lemma is a duplicate of Schemes, Lemma 26.19.5.
$\square$

Lemma 29.2.7. A closed immersion is separated.

**Proof.**
This lemma is a special case of Schemes, Lemma 26.23.8.
$\square$

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