The Stacks project

Lemma 61.19.6. Let $X$ be a scheme.

  1. 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 $i$.

  2. 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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