The Stacks project

Proof. For $X$ quasi-compact and quasi-separated, cohomology commutes with colimits, so it suffices to show the vanishing of $H_{\acute{e}tale}^ q(X, {i_ x}_* \underline{\mathbf{Z}})$. But then the inclusion $i_ x$ of a closed point is finite so $R^ p {i_ x}_* \underline{\mathbf{Z}} = 0$ for all $p \geq 1$ by Proposition 59.55.2. Applying the Leray spectral sequence, we see that $H_{\acute{e}tale}^ q(X, {i_ x}_* \underline{\mathbf{Z}}) = H_{\acute{e}tale}^ q(x, \underline{\mathbf{Z}})$. Finally, since $x$ is the spectrum of an algebraically closed field, all higher cohomology on $x$ vanishes. $\square$