Lemma 29.37.8. Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. Let $\mathcal{M}$ be an invertible $\mathcal{O}_ Y$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If $S$ is quasi-compact, $\mathcal{M}$ is $g$-ample, and $\mathcal{L}$ is $f$-ample, then $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is $g \circ f$-ample for $a \gg 0$.
Proof. Let $S = \bigcup _{i = 1, \ldots , n} V_ i$ be a finite affine open covering. By Lemma 29.37.4 it suffices to prove that $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample on $(g \circ f)^{-1}(V_ i)$ for $i = 1, \ldots , n$. Thus the lemma follows from Lemma 29.37.7. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: