The Stacks project

Lemma 38.33.5. Let $X \to S$ and $Y \to S$ be morphisms of schemes. Let $U \subset X$ be an open subscheme. Let $V \to X \times _ S Y$ be a quasi-compact morphism whose composition with the first projection maps into $U$. Let $Z \subset X \times _ S Y$ be the scheme theoretic image of $V \to X \times _ S Y$. Let $X' \to X$ be a $U$-admissible blowup. Then the scheme theoretic image of $V \to X' \times _ S Y$ is the strict transform of $Z$ with respect to the blowing up.

Proof. Denote $Z' \to Z$ the strict transform. The morphism $Z' \to X'$ induces a morphism $Z' \to X' \times _ S Y$ which is a closed immersion (as $Z'$ is a closed subscheme of $X' \times _ X Z$ by definition). Thus to finish the proof it suffices to show that the scheme theoretic image $Z''$ of $V \to Z'$ is $Z'$. Observe that $Z'' \subset Z'$ is a closed subscheme such that $V \to Z'$ factors through $Z''$. Since both $V \to X \times _ S Y$ and $V \to X' \times _ S Y$ are quasi-compact (for the latter this follows from Schemes, Lemma 26.21.14 and the fact that $X' \times _ S Y \to X \times _ S Y$ is separated as a base change of a proper morphism), by Morphisms, Lemma 29.6.3 we see that $Z \cap (U \times _ S Y) = Z'' \cap (U \times _ S Y)$. Thus the inclusion morphism $Z'' \to Z'$ is an isomorphism away from the exceptional divisor $E$ of $Z' \to Z$. However, the structure sheaf of $Z'$ does not have any nonzero sections supported on $E$ (by definition of strict transforms) and we conclude that the surjection $\mathcal{O}_{Z'} \to \mathcal{O}_{Z''}$ must be an isomorphism. $\square$

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Your email address will not be published.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 0F3Y. Beware of the difference between the letter 'O' and the digit '0'.