The Stacks project

Lemma 30.20.2. Given a morphism of schemes $f : X \to Y$, a quasi-coherent sheaf $\mathcal{F}$ on $X$, and a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Y$. Assume $Y$ locally Noetherian, $f$ proper, and $\mathcal{F}$ coherent. Then

\[ \mathcal{M} = \bigoplus \nolimits _{n \geq 0} R^ pf_*(\mathcal{I}^ n\mathcal{F}) \]

is a graded $\mathcal{A} = \bigoplus _{n \geq 0} \mathcal{I}^ n$-module which is quasi-coherent and of finite type.

Proof. The statement is local on $Y$, hence this reduces to the case where $Y$ is affine. In the affine case the result follows from Lemma 30.20.1. Details omitted. $\square$

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Your email address will not be published.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 02O9. Beware of the difference between the letter 'O' and the digit '0'.