Lemma 59.96.4 (0F0D)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.96: Finite cohomological dimension
Lemma 59.96.4 (cite)

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Lemma 59.96.4. Let $f : X \to Y$ be a proper morphism of schemes. Let $n \geq 1$ be an integer. Then the functor

$Rf_* : D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$

commutes with direct sums.

Proof. It is enough to prove this when $Y$ is quasi-compact. By Morphisms, Lemma 29.28.5 we see that the dimension of the fibres of $f : X \to Y$ is bounded. Thus Lemma 59.92.2 implies that $\text{cd}(f) < \infty$ . Hence the result by Lemma 59.96.3. $\square$

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