The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01QO. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 29.2.1 as pdf