The Stacks project

Lemma 57.9.2. Let $R$ be a ring. Let $X$, $Y$ be quasi-compact and quasi-separated schemes over $R$ having the resolution property. For any finite type quasi-coherent $\mathcal{O}_{X \times _ R Y}$-module $\mathcal{F}$ there exist a surjection $\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}$ where $\mathcal{E}$ is a finite locally free $\mathcal{O}_ X$-module and $\mathcal{G}$ is a finite locally free $\mathcal{O}_ Y$-module.

Proof. Follows from Lemma 57.9.1 by a limit argument. We urge the reader to skip the proof. Since $X \times _ R Y$ is a closed subscheme of $X \times _\mathbf {Z} Y$ it is harmless if we replace $R$ by $\mathbf{Z}$. We can write $\mathcal{F}$ as the quotient of a finitely presented $\mathcal{O}_{X \times _ R Y}$-module by Properties, Lemma 28.22.8. Hence we may assume $\mathcal{F}$ is of finite presentation. Next we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i$ of finite presentation over $\mathbf{Z}$ and similarly $Y = \mathop{\mathrm{lim}}\nolimits Y_ j$, see Limits, Proposition 32.5.4. Then $\mathcal{F}$ will descend to $\mathcal{F}_{ij}$ on some $X_ i \times _ R Y_ j$ (Limits, Lemma 32.10.2) and so does the property of having the resolution property (Derived Categories of Schemes, Lemma 36.36.9). Then we apply Lemma 57.9.1 to $\mathcal{F}_{ij}$ and we pullback. $\square$

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