The Stacks project

Lemma 56.10.1. Let $R$ be a Noetherian ring. Let $X$, $Y$ be finite type schemes over $R$ having the resolution property. For any 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. Let $U \subset X$ and $V \subset Y$ be affine open subschemes. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal sheaf of the reduced induced closed subscheme structure on $X \setminus U$. Similarly, let $\mathcal{I}' \subset \mathcal{O}_ Y$ be the ideal sheaf of the reduced induced closed subscheme structure on $Y \setminus V$. Then the ideal sheaf

\[ \mathcal{J} = \mathop{\mathrm{Im}}(\text{pr}_1^*\mathcal{I} \otimes _{\mathcal{O}_{X \times _ R Y}} \text{pr}_2^*\mathcal{I}' \to \mathcal{O}_{X \times _ R Y}) \]

satisfies $V(\mathcal{J}) = X \times _ R Y \setminus U \times _ R V$. For any section $s \in \mathcal{F}(U \times _ R V)$ we can find an integer $n > 0$ and a map $\mathcal{J}^ n \to \mathcal{F}$ whose restriction to $U \times _ R V$ gives $s$, see Cohomology of Schemes, Lemma 30.10.4. By assumption we can choose surjections $\mathcal{E} \to \mathcal{I}$ and $\mathcal{G} \to \mathcal{I}'$. These produce corresponding surjections

\[ \mathcal{E} \boxtimes \mathcal{G} \to \mathcal{J} \quad \text{and}\quad \mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{J}^ n \]

and hence a map $\mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{F}$ whose image contains the section $s$ over $U \times _ R V$. Since we can cover $X \times _ R Y$ by a finite number of affine opens of the form $U \times _ R V$ and since $\mathcal{F}|_{U \times _ R V}$ is generated by finitely many sections (Properties, Lemma 28.16.1) we conclude that there exists a surjection

\[ \bigoplus \nolimits _{j = 1, \ldots , N} \mathcal{E}_ j^{\otimes n_ j} \boxtimes \mathcal{G}_ j^{\otimes n_ j} \to \mathcal{F} \]

where $\mathcal{E}_ j$ is finite locally free on $X$ and $\mathcal{G}_ j$ is finite locally free on $Y$. Setting $\mathcal{E} = \bigoplus \mathcal{E}_ j^{\otimes n_ j}$ and $\mathcal{G} = \bigoplus \mathcal{G}_ j^{\otimes n_ j}$ we conclude that the lemma is true. $\square$


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