The Stacks project

Lemma 56.5.10. Let $R$, $X$, $Y$, and $\mathcal{K}$ be as in Lemma 56.5.7 part (2). Then for any scheme $T$ over $R$ we have

\[ R^ q\text{pr}_{13, *}(\text{pr}_{12}^*\mathcal{F} \otimes _{\mathcal{O}_{T \times _ R X \times _ R Y}} \text{pr}_{23}^*\mathcal{K}) = 0 \]

for $\mathcal{F}$ quasi-coherent on $T \times _ R X$ and $q > 0$.

Proof. The question is local on $T$ hence we may assume $T$ is affine. In this case we can consider the diagram

\[ \xymatrix{ T \times _ R X \ar[d] & T \times _ R X \times _ R Y \ar[d] \ar[l] \ar[r] & T \times _ R Y \ar[d] \\ X & X \times _ R Y \ar[l] \ar[r] & Y } \]

whose vertical arrows are affine. In particular the pushforward along $T \times _ R Y \to Y$ is faithful and exact (Cohomology of Schemes, Lemma 30.2.3 and Morphisms, Lemma 29.11.6). Chasing around in the diagram using that higher direct images along affine morphisms vanish (see reference above) we see that it suffices to prove

\[ R^ q\text{pr}_{2, *}( \text{pr}_{23, *}(\text{pr}_{12}^*\mathcal{F} \otimes _{\mathcal{O}_{T \times _ R X \times _ R Y}} \text{pr}_{23}^*\mathcal{K})) = R^ q\text{pr}_{2, *}( \text{pr}_{23, *}(\text{pr}_{12}^*\mathcal{F}) \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K})) \]

is zero which is true by assumption on $\mathcal{K}$. The equality holds by Remark 56.5.3. $\square$

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