The Stacks project

Lemma 50.12.1. Let $f : X \to Y$ be a quasi-compact, quasi-separated, and smooth morphism of schemes over a base scheme $S$. There is a bounded spectral sequence with first page

\[ E_1^{p, q} = H^ q(\Omega ^ p_{Y/S} \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\Omega ^\bullet _{X/Y}) \]

converging to $R^{p + q}f_*\Omega ^\bullet _{X/S}$.

Proof. Consider $\Omega ^\bullet _{X/S}$ as a filtered complex with the filtration introduced above. The spectral sequence is the spectral sequence of Cohomology, Lemma 20.29.5. By Derived Categories of Schemes, Lemma 36.23.2 we have

\[ Rf_*\text{gr}^ k\Omega ^\bullet _{X/S} = \Omega ^ k_{Y/S}[-k] \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\Omega ^\bullet _{X/Y} \]

and thus we conclude. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 50.12: The spectral sequence for a smooth morphism

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Your email address will not be published.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