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The Stacks project

Lemma 85.16.3. 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 ith 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 85.15.

Proof. This follows immediately from Lemma 85.8.6 and the product decompositions of Section 85.15. \square


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