The Stacks project

Lemma 59.63.1. Let $p$ be a prime. Let $S$ be a scheme of characteristic $p$.

  1. If $S$ is affine, then $H_{\acute{e}tale}^ q(S, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for all $q \geq 2$.

  2. If $S$ is a quasi-compact and quasi-separated scheme of dimension $d$, then $H_{\acute{e}tale}^ q(S, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for all $q \geq 2 + d$.

Proof. Recall that the étale cohomology of the structure sheaf is equal to its cohomology on the underlying topological space (Theorem 59.22.4). The first statement follows from the Artin-Schreier exact sequence and the vanishing of cohomology of the structure sheaf on an affine scheme (Cohomology of Schemes, Lemma 30.2.2). The second statement follows by the same argument from the vanishing of Cohomology, Proposition 20.22.4 and the fact that $S$ is a spectral space (Properties, Lemma 28.2.4). $\square$

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