The Stacks project

Lemma 29.26.1. Let $X$ be a scheme. The rule which associates to a closed subscheme of $X$ its underlying closed subset defines a bijection

\[ \left\{ \begin{matrix} \text{closed subschemes }Z \subset X \\ \text{such that }Z \to X\text{ is flat} \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{closed subsets }Z \subset X \\ \text{closed under generalizations} \end{matrix} \right\} \]

If $Z \subset X$ is such a closed subscheme, every morphism of schemes $g : Y \to X$ with $g(Y) \subset Z$ set theoretically factors (scheme theoretically) through $Z$.

Proof. The affine case of the bijection is Algebra, Lemma 10.107.4. For general schemes $X$ the bijection follows by covering $X$ by affines and glueing. Details omitted. For the final assertion, observe that the projection $Z \times _{X, g} Y \to Y$ is a flat (Lemma 29.25.8) closed immersion which is bijective on underlying topological spaces and hence must be an isomorphism by the bijection esthablished in the first part of the proof. $\square$


Comments (2)

Comment #5820 by Laurent Moret-Bailly on

Suggested addition: if is such a subscheme, every morphism mapping into factors through . (Proof: for , any section of the ideal sheaf of vanishes in ). This simplifies the proof of 0B7R, for instance.

Comment #5844 by on

Thanks for this suggestion. I have made the corresponding changes here.


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Your email address will not be published.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 04PW. Beware of the difference between the letter 'O' and the digit '0'.