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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