The Stacks project

Proof. Choose a scheme $W$ and a surjective étale morphism $W \to B$. Choose a scheme $U$ and a surjective étale morphism $U \to X \times _ B W$. Choose a scheme $V$ and a surjective étale morphism $V \to Y \times _ B W$. Then $U \times _ W V \to X \times _ B Y$ is surjective étale too. Hence it suffices to prove that the restriction of $E$ to $U \times _ W V$ is zero. By Lemma 92.26.3 and Derived Categories of Spaces, Lemma 75.20.3 this reduces us to the case of schemes. Taking suitable affine opens we reduce to the case of affine schemes. Using Lemma 92.24.2 we reduce to the case of a tensor product of rings, i.e., to Lemma 92.15.1. $\square$