The Stacks project

Lemma 85.11.2. With notation as above. For a $\mathcal{O}_\mathcal {D}$-module $\mathcal{G}$ there is an exact complex

\[ \ldots \to g_{2!}(a_2^*\mathcal{G}) \to g_{1!}(a_1^*\mathcal{G}) \to g_{0!}(a_0^*\mathcal{G}) \to a^*\mathcal{G} \to 0 \]

of sheaves of $\mathcal{O}$-modules on $\mathcal{C}_{total}$. Here $g_{n!}$ is as in Lemma 85.6.1.

Proof. Observe that $a^*\mathcal{G}$ is the $\mathcal{O}$-module on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ m$ is the $\mathcal{O}_ m$-module $a_ m^*\mathcal{G}$. The description of the functors $g_{n!}$ on modules in Lemma 85.6.1 shows that $g_{n!}(a_ n^*\mathcal{G})$ is the $\mathcal{O}$-module on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ m$ is the $\mathcal{O}_ m$-module

\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^*a_ n^*\mathcal{G} = \bigoplus \nolimits _{\varphi : [n] \to [m]} a_ m^*\mathcal{G} \]

The rest of the proof is exactly the same as the proof of Lemma 85.9.1, replacing $a_ m^{-1}\mathcal{G}$ by $a_ m^*\mathcal{G}$. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 85.11: Cohomology and augmentations of ringed simplicial sites

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Your email address will not be published.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 0D7C. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 85.11.2 as pdf