Proper base change for étale cohomology holds for finite morphisms.

Lemma 57.87.9. Let $f : X \to Y$ be a finite morphism of schemes. Then cohomology commutes with base change for $f$.

Proof. Observe that a finite morphism is proper, see Morphisms, Lemma 29.42.11. Moreover, the base change of a finite morphism is finite, see Morphisms, Lemma 29.42.6. Thus the result follows from Lemma 57.87.6 combined with Proposition 57.54.2. $\square$

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Suggested slogan: Etale cohomology commutes with finite base change.

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