The Stacks project

Lemma 35.23.19. The property $\mathcal{P}(f) =$“$f$ is a closed immersion” is fpqc local on the base.

Proof. Let $f : X \to Y$ be a morphism of schemes. Let $\{ Y_ i \to Y\} $ be an fpqc covering. Assume that each $f_ i : Y_ i \times _ Y X \to Y_ i$ is a closed immersion. This implies that each $f_ i$ is affine, see Morphisms, Lemma 29.11.9. By Lemma 35.23.18 we conclude that $f$ is affine. It remains to show that $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is surjective. For every $y \in Y$ there exists an $i$ and a point $y_ i \in Y_ i$ mapping to $y$. By Cohomology of Schemes, Lemma 30.5.2 the sheaf $f_{i, *}(\mathcal{O}_{Y_ i \times _ Y X})$ is the pullback of $f_*\mathcal{O}_ X$. By assumption it is a quotient of $\mathcal{O}_{Y_ i}$. Hence we see that

\[ \Big( \mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_ X)_ y \Big) \otimes _{\mathcal{O}_{Y, y}} \mathcal{O}_{Y_ i, y_ i} \]

is surjective. Since $\mathcal{O}_{Y_ i, y_ i}$ is faithfully flat over $\mathcal{O}_{Y, y}$ this implies the surjectivity of $\mathcal{O}_{Y, y} \longrightarrow (f_*\mathcal{O}_ X)_ y$ as desired. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 35.23: Properties of morphisms local in the fpqc topology on the target

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Your email address will not be published.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 02L6. Beware of the difference between the letter 'O' and the digit '0'.