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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