The Stacks project

[IV Corollary 9.6.4, EGA]

Lemma 37.49.3. Let $f : X \to Y$ be a proper morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $y \in Y$ be a point such that $\mathcal{L}_ y$ is ample on $X_ y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.

Proof. We may assume $Y$ is affine. Then we find a directed set $I$ and an inverse system of morphisms $X_ i \to Y_ i$ of schemes with $Y_ i$ of finite type over $\mathbf{Z}$, with affine transition morphisms $X_ i \to X_{i'}$ and $Y_ i \to Y_{i'}$, with $X_ i \to Y_ i$ proper, such that $X \to Y = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$. See Limits, Lemma 32.13.3. After shrinking $I$ we can assume we have a compatible system of invertible $\mathcal{O}_{X_ i}$-modules $\mathcal{L}_ i$ pulling back to $\mathcal{L}$, see Limits, Lemma 32.10.3. Let $y_ i \in Y_ i$ be the image of $y$. Then $\kappa (y) = \mathop{\mathrm{colim}}\nolimits \kappa (y_ i)$. Hence for some $i$ we have $\mathcal{L}_{i, y_ i}$ is ample on $X_{i, y_ i}$ by Limits, Lemma 32.4.15. By Cohomology of Schemes, Lemma 30.21.4 we find an open neigbourhood $V_ i \subset Y_ i$ of $y_ i$ such that $\mathcal{L}_ i$ restricted to $f_ i^{-1}(V_ i)$ is ample relative to $V_ i$. Letting $V \subset Y$ be the inverse image of $V_ i$ finishes the proof (hints: use Morphisms, Lemma 29.37.9 and the fact that $X \to Y \times _{Y_ i} X_ i$ is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma 29.37.7). $\square$


Comments (2)

Comment #2688 by on

A reference is EGA IV_3, Corollary 9.6.4


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Your email address will not be published.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 0D2S. Beware of the difference between the letter 'O' and the digit '0'.