Lemma 85.18.2. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering. For $E \in D(\mathcal{O}_\mathcal {C})$ the map
is an isomorphism.
Lemma 85.18.2. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering. For $E \in D(\mathcal{O}_\mathcal {C})$ the map
is an isomorphism.
Proof. Since $a^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}$ we have $La^* = a^* = a^{-1}$. Moreover $Ra_*$ agrees with $Ra_*$ on abelian sheaves, see Cohomology on Sites, Lemma 21.20.7. Hence the lemma follows immediately from Lemma 85.17.3. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)