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

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

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. $\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)