The Stacks project

Remark 85.17.6. Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be a presheaf of sets on $\mathcal{C}$. If $\mathcal{C}$ has equalizers and fibre products, then we've defined the notion of a hypercovering of $\mathcal{G}$ in Hypercoverings, Definition 25.6.1. We claim that all the results in this section have a valid counterpart in this setting. To see this, define the localization $\mathcal{C}/\mathcal{G}$ of $\mathcal{C}$ at $\mathcal{G}$ exactly as in Sites, Lemma 7.30.3 (which is stated only for sheaves; the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/\mathcal{G})$ is equal to the localization of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ at the sheaf $\mathcal{G}^\# $). Then the reader easily shows that the site $\mathcal{C}/\mathcal{G}$ has fibre products and equalizers and that a hypercovering of $\mathcal{G}$ in $\mathcal{C}$ is the same thing as a hypercovering for the site $\mathcal{C}/\mathcal{G}$. Hence replacing the site $\mathcal{C}$ by $\mathcal{C}/\mathcal{G}$ in the lemmas on hypercoverings above we obtain proofs of the corresponding results for hypercoverings of $\mathcal{G}$. Example: for a hypercovering $K$ of $\mathcal{G}$ we have

\[ R\Gamma (\mathcal{C}/\mathcal{G}, E) = R\Gamma ((\mathcal{C}/K)_{total}, a^{-1}E) \]

for $E \in D^+(\mathcal{C}/\mathcal{G})$ where $a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/\mathcal{G})$ is the canonical augmentation. This is Lemma 85.17.4. Let $R\Gamma (\mathcal{G}, -) : D(\mathcal{C}) \to D(\textit{Ab})$ be defined as the derived functor of the functor $H^0(\mathcal{G}, -) = H^0(\mathcal{G}^\# , -)$ discussed in Hypercoverings, Section 25.6 and Cohomology on Sites, Section 21.13. We have

\[ R\Gamma (\mathcal{G}, E) = R\Gamma (\mathcal{C}/\mathcal{G}, j^{-1}E) \]

by the analogue of Cohomology on Sites, Lemma 21.7.1 for the localization fuctor $j : \mathcal{C}/\mathcal{G} \to \mathcal{C}$. Putting everything together we obtain

\[ R\Gamma (\mathcal{G}, E) = R\Gamma ((\mathcal{C}/K)_{total}, a^{-1}j^{-1}E) = R\Gamma ((\mathcal{C}/K)_{total}, g^{-1}E) \]

for $E \in D^+(\mathcal{C})$ where $g : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is the composition of $a$ and $j$.

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Your email address will not be published.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 09X6. Beware of the difference between the letter 'O' and the digit '0'.