The Stacks project

Remark 20.13.2. Here is a down-to-earth explanation of the meaning of Lemma 20.13.1. It says that given $f : X \to Y$ and $\mathcal{F} \in \textit{Mod}(\mathcal{O}_ X)$ and given an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ we have

\[ \begin{matrix} R\Gamma (X, \mathcal{F}) & \text{is represented by} & \Gamma (X, \mathcal{I}^\bullet ) \\ Rf_*\mathcal{F} & \text{is represented by} & f_*\mathcal{I}^\bullet \\ R\Gamma (Y, Rf_*\mathcal{F}) & \text{is represented by} & \Gamma (Y, f_*\mathcal{I}^\bullet ) \end{matrix} \]

the last fact coming from Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) and Lemma 20.11.10. Finally, it combines this with the trivial observation that

\[ \Gamma (X, \mathcal{I}^\bullet ) = \Gamma (Y, f_*\mathcal{I}^\bullet ). \]

to arrive at the commutativity of the diagram of the lemma.

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Your email address will not be published.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 01F0. Beware of the difference between the letter 'O' and the digit '0'.