The Stacks project

Lemma 21.20.5. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Then $R\Gamma (\mathcal{D}, -) \circ Rf_* = R\Gamma (\mathcal{C}, -)$ as functors $D(\mathcal{O}_\mathcal {C}) \to D(\Gamma (\mathcal{O}_\mathcal {D}))$. More generally, for $V \in \mathcal{D}$ with $U = u(V)$ we have $R\Gamma (U, -) = R\Gamma (V, -) \circ Rf_*$.

Proof. Consider the punctual topos $pt$ endowed with $\mathcal{O}_{pt}$ given by the ring $\Gamma (\mathcal{O}_\mathcal {D})$. There is a canonical morphism $(\mathcal{D}, \mathcal{O}_\mathcal {D}) \to (pt, \mathcal{O}_{pt})$ of ringed topoi inducing the identification on global sections of structure sheaves. Then $D(\mathcal{O}_{pt}) = D(\Gamma (\mathcal{O}_\mathcal {D}))$. The assertion $R\Gamma (\mathcal{D}, -) \circ Rf_* = R\Gamma (\mathcal{C}, -)$ follows from Lemma 21.19.2 applied to

\[ (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D}) \to (pt, \mathcal{O}_{pt}) \]

The second (more general) statement follows from the first statement after applying Lemma 21.20.4. $\square$

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Your email address will not be published.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 0D6H. Beware of the difference between the letter 'O' and the digit '0'.