The Stacks project

Remark 48.28.8. Let $X \to S$ be a morphism of schemes which is flat, proper, and of finite presentation. By Lemma 48.28.5 there exists a relative dualizing complex $(\omega _{X/S}^\bullet , \xi )$ in the sense of Definition 48.28.1. Consider any morphism $g : S' \to S$ where $S'$ is quasi-compact and quasi-separated (for example an affine open of $S$). By Lemma 48.28.6 we see that $(L(g')^*\omega _{X/S}^\bullet , L(g')^*\xi )$ is a relative dualizing complex for the base change $f' : X' \to S'$ in the sense of Definition 48.28.1. Let $\omega _{X'/S'}^\bullet $ be the relative dualizing complex for $X' \to S'$ in the sense of Remark 48.12.5. Combining Lemmas 48.28.7 and 48.28.4 we see that there is a unique isomorphism

\[ \omega _{X'/S'}^\bullet \longrightarrow L(g')^*\omega _{X/S}^\bullet \]

compatible with ( and $L(g')^*\xi $. These isomorphisms are compatible with morphisms between quasi-compact and quasi-separated schemes over $S$ and the base change isomorphisms of Lemma 48.12.4 (if we ever need this compatibility we will carefully state and prove it here).

Comments (0)

There are also:

  • 2 comment(s) on Section 48.28: Relative dualizing complexes

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Your email address will not be published.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 0E4P. Beware of the difference between the letter 'O' and the digit '0'.