The Stacks project

Lemma 37.30.7. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes satisfying the hypotheses of Lemma 37.30.1. 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 K√ľnneth (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 37.30.6 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. $\square$


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