The Stacks project

[II, Proposition 4.6.2, EGA]

Lemma 29.38.2. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If $f$ is quasi-compact and $\mathcal{L}$ is a relatively very ample invertible sheaf, then $\mathcal{L}$ is a relatively ample invertible sheaf.

Proof. By definition there exists quasi-coherent $\mathcal{O}_ S$-module $\mathcal{E}$ and an immersion $i : X \to \mathbf{P}(\mathcal{E})$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. Set $\mathcal{A} = \text{Sym}(\mathcal{E})$, so $\mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_ S(\mathcal{A})$ by definition. The graded $\mathcal{O}_ S$-algebra $\mathcal{A}$ comes equipped with a map

\[ \psi : \mathcal{A} \to \bigoplus \nolimits _{n \geq 0} \pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n) \to \bigoplus \nolimits _{n \geq 0} f_*\mathcal{L}^{\otimes n} \]

where the second arrow uses the identification $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. By adjointness of $f_*$ and $f^*$ we get a morphism $\psi : f^*\mathcal{A} \to \bigoplus _{n \geq 0}\mathcal{L}^{\otimes n}$. We omit the verification that the morphism $r_{\mathcal{L}, \psi }$ associated to this map is exactly the immersion $i$. Hence the result follows from part (6) of Lemma 29.37.4. $\square$


Comments (2)

Comment #2723 by Matt Stevenson on

This is EGA II Prop 4.6.2.


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Your email address will not be published.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 01VN. Beware of the difference between the letter 'O' and the digit '0'.