Lemma 46.7.4. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. For any object $\mathcal{F}$ of $\textit{Adeq}(\mathcal{O})$ we have $R^ pA(\mathcal{F}) = 0$ for all $p > 0$ where $A$ is as in Lemma 46.7.3.
Proof. With notation as in the proof of Lemma 46.7.3 choose an injective resolution $k(\mathcal{F}) \to \mathcal{I}^\bullet $ in the category of $\mathcal{O}$-modules on $(\textit{Aff}/U)_\tau $. By Cohomology on Sites, Lemmas 21.12.2 and Lemma 46.5.8 the complex $j(\mathcal{I}^\bullet )$ is exact. On the other hand, each $j(\mathcal{I}^ n)$ is an injective object of the category of presheaves of modules by Cohomology on Sites, Lemma 21.12.1. It follows that $R^ pA(\mathcal{F}) = R^ pQ(j(k(\mathcal{F})))$. Hence the result now follows from Lemma 46.4.10. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)