Lemma 37.32.7. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes satisfying the hypotheses of Lemma 37.32.1. Let $\sigma : S \to X$ and $\tau : S \to Y$ be sections of $f$ and $g$. Let $s \in S$. Let $\mathcal{L}$ be an invertible sheaf on $X \times _ S Y$. If $(1 \times \tau )^*\mathcal{L}$ on $X$, $(\sigma \times 1)^*\mathcal{L}$ on $Y$, and $\mathcal{L}|_{(X \times _ S Y)_ s}$ are trivial, then there is an open neighbourhood $U$ of $s$ such that $\mathcal{L}$ is trivial over $(X \times _ S Y)_ U$.

**Proof.**
By Künneth (Varieties, Lemma 33.29.1) the map

\[ H^1(X_ s \times _{\mathop{\mathrm{Spec}}(\kappa (s))} Y_ s, \mathcal{O}) \to H^1(X_ s, \mathcal{O}) \oplus H^1(Y_ s, \mathcal{O}) \]

is injective. Thus we may apply Lemma 37.32.6 to the two morphisms

\[ 1 \times \tau : X \to X \times _ S Y \quad \text{and}\quad \sigma \times 1 : Y \to X \times _ S Y \]

to conclude. $\square$

## Post a comment

Your email address will not be published.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)