The Stacks project

Lemma 56.9.2. Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times _ S Y})$. The corresponding Fourier-Mukai functor $\Phi _ K$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$ if $K$ is in $D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$ and $X \to S$ is quasi-compact and quasi-separated.

Proof. This follows from the fact that derived pullback preserves $D_\mathit{QCoh}$ (Derived Categories of Schemes, Lemma 36.3.8), derived tensor products preserve $D_\mathit{QCoh}$ (Derived Categories of Schemes, Lemma 36.3.9), the projection $\text{pr}_2 : X \times _ S Y \to Y$ is quasi-compact and quasi-separated (Schemes, Lemmas 26.19.3 and 26.21.12), and total direct image along a quasi-separated and quasi-compact morphism preserves $D_\mathit{QCoh}$ (Derived Categories of Schemes, Lemma 36.4.1). $\square$


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Your email address will not be published.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 0FYR. Beware of the difference between the letter 'O' and the digit '0'.