The Stacks project

Lemma 28.26.15. Let $S$ be a quasi-separated scheme. Let $X$, $Y$ be schemes over $S$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module and let $\mathcal{N}$ be an ample invertible $\mathcal{O}_ Y$-module. Then $\mathcal{M} = \text{pr}_1^*\mathcal{L} \otimes _{\mathcal{O}_{X \times _ S Y}} \text{pr}_2^*\mathcal{N}$ is an ample invertible sheaf on $X \times _ S Y$.

Proof. The morphism $i : X \times _ S Y \to X \times Y$ is a quasi-compact immersion, see Schemes, Lemma 26.21.9. On the other hand, $\mathcal{M}$ is the pullback by $i$ of the corresponding invertible module on $X \times Y$. By Lemma 28.26.14 it suffices to prove the lemma for $X \times Y$. We check (1) and (2) of Definition 28.26.1 for $\mathcal{M}$ on $X \times Y$.

Since $X$ and $Y$ are quasi-compact, so is $X \times Y$. Let $z \in X \times Y$ be a point. Let $x \in X$ and $y \in Y$ be the projections. Choose $n > 0$ and $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ such that $X_ s$ is an affine open neighbourhood of $x$. Choose $m > 0$ and $t \in \Gamma (Y, \mathcal{N}^{\otimes m})$ such that $Y_ t$ is an affine open neighbourhood of $y$. Then $r = \text{pr}_1^*s \otimes \text{pr}_2^*t$ is a section of $\mathcal{M}$ with $(X \times Y)_ r = X_ s \times Y_ t$. This is an affine open neighbourhood of $z$ and the proof is complete. $\square$

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