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). $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).