Lemma 85.25.4 (09XC)—The Stacks project

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Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.25: Proper hypercoverings in topology
Lemma 85.25.4 (cite)

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Lemma 85.25.4. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $U$ is a proper hypercovering of $X$ , then

$R\Gamma (X, K) = R\Gamma (U_{Zar}, a^{-1}K)$

for $K \in D^+(X)$ where $a : \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ is as in Lemma 85.2.8.

Proof. This follows from Lemma 85.25.3 because $R\Gamma (U_{Zar}, -) = R\Gamma (X, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4. $\square$

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