Lemma 85.8.6. Let $\mathcal{C}$ be as in Situation 85.3.3. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ n)$. Let $\mathcal{F} \in \textit{Ab}(\mathcal{C}_{total})$. Then $H^ p(U, \mathcal{F}) = H^ p(U, g_ n^{-1}\mathcal{F})$ 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). The equality then follows from Lemma 85.3.6 and the definition of cohomology. $\square$
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