The Stacks project

Remark 21.39.4. Notation and assumptions as in Example 21.39.1. Let $\mathcal{F}^\bullet $ be a bounded complex of abelian sheaves on $\mathcal{C}$. For any object $U$ of $\mathcal{C}$ there is a canonical map

\[ \mathcal{F}^\bullet (U) \longrightarrow L\pi _!(\mathcal{F}^\bullet ) \]

in $D(\textit{Ab})$. If $\mathcal{F}^\bullet $ is a complex of $\underline{B}$-modules then this map is in $D(B)$. To prove this, note that we compute $L\pi _!(\mathcal{F}^\bullet )$ by taking a quasi-isomorphism $\mathcal{P}^\bullet \to \mathcal{F}^\bullet $ where $\mathcal{P}^\bullet $ is a complex of projectives. However, since the topology is chaotic this means that $\mathcal{P}^\bullet (U) \to \mathcal{F}^\bullet (U)$ is a quasi-isomorphism hence can be inverted in $D(\textit{Ab})$, resp. $D(B)$. Composing with the canonical map $\mathcal{P}^\bullet (U) \to \pi _!(\mathcal{P}^\bullet )$ coming from the computation of $\pi _!$ as a colimit we obtain the desired arrow.

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Your email address will not be published.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 08Q6. Beware of the difference between the letter 'O' and the digit '0'.