The Stacks project

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:

  1. The morphism $i$ is a closed immersion.

  2. 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 $U = \mathop{\mathrm{Spec}}(R)$.

  3. 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 $U_ j = \mathop{\mathrm{Spec}}(R_ j)$.

  4. 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.

  5. 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.

  6. 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 $\mathcal{O}_ X$.

  1. 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.

  2. 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:

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

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

  3. 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 $(R/I)/\overline{J} = R/J$. $\square$

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

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

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