The Stacks project

Lemma 29.37.7. Let $f : X \to Y$ be a morphism of schemes, $\mathcal{M}$ an invertible $\mathcal{O}_ Y$-module, and $\mathcal{L}$ an invertible $\mathcal{O}_ X$-module.

  1. If $\mathcal{L}$ is $f$-ample and $\mathcal{M}$ is ample, then $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample for $a \gg 0$.

  2. If $\mathcal{M}$ is ample and $f$ quasi-affine, then $f^*\mathcal{M}$ is ample.

Proof. Assume $\mathcal{L}$ is $f$-ample and $\mathcal{M}$ ample. By assumption $Y$ and $f$ are quasi-compact (see Definition 29.37.1 and Properties, Definition 28.26.1). Hence $X$ is quasi-compact. By Properties, Lemma 28.26.8 the scheme $Y$ is separated and by Lemma 29.37.3 the morphism $f$ is separated. Hence $X$ is separated by Schemes, Lemma 26.21.12. Pick $x \in X$. We can choose $m \geq 1$ and $t \in \Gamma (Y, \mathcal{M}^{\otimes m})$ such that $Y_ t$ is affine and $f(x) \in Y_ t$. Since $\mathcal{L}$ restricts to an ample invertible sheaf on $f^{-1}(Y_ t) = X_{f^*t}$ we can choose $n \geq 1$ and $s \in \Gamma (X_{f^*t}, \mathcal{L}^{\otimes n})$ with $x \in (X_{f^*t})_ s$ with $(X_{f^*t})_ s$ affine. By Properties, Lemma 28.17.2 part (2) whose assumptions are satisfied by the above, there exists an integer $e \geq 1$ and a section $s' \in \Gamma (X, \mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes em})$ which restricts to $s(f^*t)^ e$ on $X_{f^*t}$. For any $b > 0$ consider the section $s'' = s'(f^*t)^ b$ of $\mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes (e + b)m}$. Then $X_{s''} = (X_{f^*t})_ s$ is an affine open of $X$ containing $x$. Picking $b$ such that $n$ divides $e + b$ we see $\mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes (e + b)m}$ is the $n$th power of $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ for some $a$ and we can get any $a$ divisible by $m$ and big enough. Since $X$ is quasi-compact a finite number of these affine opens cover $X$. We conclude that for some $a$ sufficiently divisible and large enough the invertible sheaf $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample on $X$. On the other hand, we know that $\mathcal{M}^{\otimes c}$ (and hence its pullback to $X$) is globally generated for all $c \gg 0$ by Properties, Proposition 28.26.13. Thus $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a + c}$ is ample (Properties, Lemma 28.26.5) for $c \gg 0$ and (1) is proved.

Part (2) follows from Lemma 29.37.6, Properties, Lemma 28.26.2, and part (1). $\square$


Comments (0)

There are also:

  • 3 comment(s) on Section 29.37: Relatively ample sheaves

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 0892. Beware of the difference between the letter 'O' and the digit '0'.