The Stacks project

Lemma 37.61.8. Let $i : Z \to X$ and $j : Z \to Y$ be closed immersions of schemes. Then the pushout $Y \amalg _ Z X$ exists in the category of schemes. Picture

\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ j & X \ar[d]^ a \\ Y \ar[r]^-b & Y \amalg _ Z X } \]

The diagram is a fibre square, the morphisms $a$ and $b$ are closed immersions, and there is a short exact sequence

\[ 0 \to \mathcal{O}_{Y \amalg _ Z X} \to a_*\mathcal{O}_ X \oplus b_*\mathcal{O}_ Y \to c_*\mathcal{O}_ Z \to 0 \]

where $c = a \circ i = b \circ j$.

Proof. This is a special case of Proposition 37.61.3. Observe that hypothesis (3) in Situation 37.61.1 is immediate because the fibres of $j$ are singletons. Finally, reverse the roles of the arrows to conclude that both $a$ and $b$ are closed immersions. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 37.61: Pushouts in the category of schemes, II

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Your email address will not be published.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 0B7M. Beware of the difference between the letter 'O' and the digit '0'.