The Stacks project

Remark 48.20.13. Let $f : X \to Y$ be a flat proper morphism of finite type schemes over a pair $(S, \omega _ S^\bullet )$ as in Situation 48.20.1. The relative dualizing complex (Remark 48.12.5) is $\omega _{X/Y}^\bullet = a(\mathcal{O}_ Y)$. By Lemma 48.20.8 we have the first canonical isomorphism in

\[ \omega _ X^\bullet = a(\omega _ Y^\bullet ) = Lf^*\omega _ Y^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}^\bullet \]

in $D(\mathcal{O}_ X)$. The second canonical isomorphism follows from the discussion in Remark 48.12.5.

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Your email address will not be published.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 0B6W. Beware of the difference between the letter 'O' and the digit '0'.