The Stacks project

Lemma 84.16.3. Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\text{SR}(\mathcal{C})$. Let $U/U_{n, i}$ be an object of $\mathcal{C}/K_ n$. Let $\mathcal{F} \in \textit{Ab}((\mathcal{C}/K)_{total})$. Then

\[ H^ p(U, \mathcal{F}) = H^ p(U, \mathcal{F}_{n, i}) \]

where

  1. on the left hand side $U$ is viewed as an object of $\mathcal{C}_{total}$, and

  2. on the right hand side $\mathcal{F}_{n, i}$ is the $i$th component of the sheaf $\mathcal{F}_ n$ on $\mathcal{C}/K_ n$ in the decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n) = \prod \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_{n, i})$ of Section 84.15.

Proof. This follows immediately from Lemma 84.8.6 and the product decompositions of Section 84.15. $\square$

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