The Stacks project

Proposition 59.45.4 (Topological invariance of étale cohomology). Let $X_0 \to X$ be a universal homeomorphism of schemes (for example the closed immersion defined by a nilpotent sheaf of ideals). Then

  1. the étale sites $X_{\acute{e}tale}$ and $(X_0)_{\acute{e}tale}$ are isomorphic,

  2. the étale topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ and $\mathop{\mathit{Sh}}\nolimits ((X_0)_{\acute{e}tale})$ are equivalent, and

  3. $H^ q_{\acute{e}tale}(X, \mathcal{F}) = H^ q_{\acute{e}tale}(X_0, \mathcal{F}|_{X_0})$ for all $q$ and for any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$.

Proof. The equivalence of categories $X_{\acute{e}tale}\to (X_0)_{\acute{e}tale}$ is given by Theorem 59.45.2. We omit the proof that under this equivalence the étale coverings correspond. Hence (1) holds. Parts (2) and (3) follow formally from (1). $\square$

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