Lemma 85.10.4. In Situation 85.3.3 let $\mathcal{O}$ be a sheaf of rings. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ n)$. Let $\mathcal{F} \in \textit{Mod}(\mathcal{O})$. Then $H^ p(U, \mathcal{F}) = H^ p(U, g_ n^*\mathcal{F})$ where on the left hand side $U$ is viewed as an object of $\mathcal{C}_{total}$.
Proof. Observe that “$U$ viewed as object of $\mathcal{C}_{total}$” is explained by the construction of $\mathcal{C}_{total}$ in Lemma 85.3.1 in case (A) and Lemma 85.3.2 in case (B). In both cases the functor $\mathcal{C}_ n \to \mathcal{C}$ is continuous and cocontinuous, see Lemma 85.3.5, and $g_ n^{-1}\mathcal{O} = \mathcal{O}_ n$ by definition. Hence the result is a special case of Cohomology on Sites, Lemma 21.37.5. $\square$
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