The Stacks project

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. $\square$

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