The Stacks project

Blowing up commutes with flat base change.

Lemma 31.32.3. Let $X_1 \to X_2$ be a flat morphism of schemes. Let $Z_2 \subset X_2$ be a closed subscheme. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_ i$ be the blowup of $Z_ i$ in $X_ i$. Then there exists a cartesian diagram

\[ \xymatrix{ X_1' \ar[r] \ar[d] & X_2' \ar[d] \\ X_1 \ar[r] & X_2 } \]

of schemes.

Proof. Let $\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$. Denote $g : X_1 \to X_2$ the given morphism. Then the ideal sheaf $\mathcal{I}_1$ of $Z_1$ is the image of $g^*\mathcal{I}_2 \to \mathcal{O}_{X_1}$ (by definition of the inverse image, see Schemes, Definition 26.17.7). By Constructions, Lemma 27.16.10 we see that $X_1 \times _{X_2} X_2'$ is the relative Proj of $\bigoplus _{n \geq 0} g^*\mathcal{I}_2^ n$. Because $g$ is flat the map $g^*\mathcal{I}_2^ n \to \mathcal{O}_{X_1}$ is injective with image $\mathcal{I}_1^ n$. Thus we see that $X_1 \times _{X_2} X_2' = X_1'$. $\square$

Comments (1)

Comment #3600 by slogan_bot on

Suggested slogan: Blowing up commutes with flat base change.

There are also:

  • 7 comment(s) on Section 31.32: Blowing up

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Your email address will not be published.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 0805. Beware of the difference between the letter 'O' and the digit '0'.