The Stacks project

Lemma 92.28.1. In the situation above, if $X$ and $Y$ are Tor independent over $B$, then the object $E$ in ( is zero. In this case we have

\[ L_{X \times _ B Y/B} = Lp^*L_{X/B} \oplus Lq^*L_{Y/B} \]

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. $\square$

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