The Stacks project

Remark 92.25.2. In the situation above, for every $U \subset X$ open let $P_{\bullet , U}$ be the standard resolution of $\mathcal{O}_ X(U)$ over $\Lambda $. Set $\mathbf{A}_{n, U} = \mathop{\mathrm{Spec}}(P_{n, U})$. Then $\mathbf{A}_{\bullet , U}$ is a cosimplicial object of the fibre category $\mathcal{C}_{\mathcal{O}_ X(U)/\Lambda }$ of $\mathcal{C}_{X/\Lambda }$ over $U$. Moreover, as discussed in Remark 92.5.5 we have that $\mathbf{A}_{\bullet , U}$ is a cosimplicial object of $\mathcal{C}_{\mathcal{O}_ X(U)/\Lambda }$ as in Cohomology on Sites, Lemma 21.39.7. Since the construction $U \mapsto \mathbf{A}_{\bullet , U}$ is functorial in $U$, given any (abelian) sheaf $\mathcal{F}$ on $\mathcal{C}_{X/\Lambda }$ we obtain a complex of presheaves

\[ U \longmapsto \mathcal{F}(\mathbf{A}_{\bullet , U}) \]

whose cohomology groups compute the homology of $\mathcal{F}$ on the fibre category. We conclude by Cohomology on Sites, Lemma 21.40.2 that the sheafification computes $L_ n\pi _!(\mathcal{F})$. In other words, the complex of sheaves whose term in degree $-n$ is the sheafification of $U \mapsto \mathcal{F}(\mathbf{A}_{n, U})$ computes $L\pi _!(\mathcal{F})$.


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Your email address will not be published.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