The Stacks project

Lemma 57.10.2. 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 57.10.1 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 57.10.1 to $\mathcal{F}_{ij}$ and we pullback. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FZ1. Beware of the difference between the letter 'O' and the digit '0'.