The Stacks project

Lemma 29.3.1. Let $Z \to Y \to X$ be morphisms of schemes.

  1. If $Z \to X$ is an immersion, then $Z \to Y$ is an immersion.

  2. If $Z \to X$ is a quasi-compact immersion and $Y \to X$ is quasi-separated, then $Z \to Y$ is a quasi-compact immersion.

  3. If $Z \to X$ is a closed immersion and $Y \to X$ is separated, then $Z \to Y$ is a closed immersion.

Proof. In each case the proof is to contemplate the commutative diagram

\[ \xymatrix{ Z \ar[r] \ar[rd] & Y \times _ X Z \ar[r] \ar[d] & Z \ar[d] \\ & Y \ar[r] & X } \]

where the composition of the top horizontal arrows is the identity. Let us prove (1). The first horizontal arrow is a section of $Y \times _ X Z \to Z$, whence an immersion by Schemes, Lemma 26.21.11. The arrow $Y \times _ X Z \to Y$ is a base change of $Z \to X$ hence an immersion (Schemes, Lemma 26.18.2). Finally, a composition of immersions is an immersion (Schemes, Lemma 26.24.3). This proves (1). The other two results are proved in exactly the same manner. $\square$

Comments (4)

Comment #733 by Kestutis Cesnavicius on

Is the analogue of 3) for algebraic spaces stated anywhere in the SP? (The proof is the same and based on ). Also, it seems to me that section 28.2. would be a more natural place for part 3).

Comment #739 by on

OK, I added a similar lemma to stacks-morphisms.tex You can find it in this commit. It will appear on the web soonish. Thanks!

Comment #4342 by Kazuki Masugi(馬杉和貴) on

in (2), " is quasi-separated" should be " is quasi-separated".

There are also:

  • 4 comment(s) on Section 29.3: Immersions

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Your email address will not be published.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 07RK. Beware of the difference between the letter 'O' and the digit '0'.