Lemma 26.21.9. Let $f : X \to T$ and $g : Y \to T$ be morphisms of schemes with the same target. Let $h : T \to S$ be a morphism of schemes. Then the induced morphism $i : X \times _ T Y \to X \times _ S Y$ is an immersion. If $T \to S$ is separated, then $i$ is a closed immersion. If $T \to S$ is quasi-separated, then $i$ is a quasi-compact morphism.

**Proof.**
By general category theory the following diagram

\[ \xymatrix{ X \times _ T Y \ar[r] \ar[d] & X \times _ S Y \ar[d] \\ T \ar[r]^{\Delta _{T/S}} \ar[r] & T \times _ S T } \]

is a fibre product diagram. The lemma follows from Lemmas 26.21.2, 26.17.6 and 26.19.3. $\square$

## 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Your email address will not be published.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.

## Comments (2)

Comment #8468 by ElĂas Guisado on

Comment #9083 by Stacks project on

There are also: