Lemma 61.19.6 (099W)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.19: Comparison with the étale site
Lemma 61.19.6 (cite)

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Lemma 61.19.6. Let $X$ be a scheme.

For an abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have

$H^ i(X_{\acute{e}tale}, \mathcal{F}) = H^ i(X_{pro\text{-}\acute{e}tale}, \epsilon ^{-1}\mathcal{F})$

for all $i$ .
For $K \in D^+(X_{\acute{e}tale})$ we have

$R\Gamma (X_{\acute{e}tale}, K) = R\Gamma (X_{pro\text{-}\acute{e}tale}, \epsilon ^{-1}K)$

Proof. Immediate consequence of Lemma 61.19.5 and the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.6). $\square$

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