The Stacks project

Lemma 68.20.1. Let $S$ be a scheme. Consider a commutative diagram

\[ \xymatrix{ X \ar[r]_ i \ar[rd]_ f & \mathbf{P}^ n_ Y \ar[d] \\ & Y } \]

of algebraic spaces over $S$. Assume $i$ is a closed immersion and $Y$ Noetherian. Set $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n_ Y}(1)$. Let $\mathcal{F}$ be a coherent module on $X$. Then there exists an integer $d_0$ such that for all $d \geq d_0$ we have $R^ pf_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$.

Proof. Checking whether $R^ pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes d})$ is zero can be done ├ętale locally on $Y$, see Equation ( Hence we may assume $Y$ is the spectrum of a Noetherian ring. In this case $X$ is a scheme and the result follows from Cohomology of Schemes, Lemma 30.16.2. $\square$

Comments (0)

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Your email address will not be published.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 08AQ. Beware of the difference between the letter 'O' and the digit '0'.