Remark 20.13.5. The Leray spectral sequence, the way we proved it in Lemma 20.13.4 is a spectral sequence of $\Gamma (Y, \mathcal{O}_ Y)$-modules. However, it is quite easy to see that it is in fact a spectral sequence of $\Gamma (X, \mathcal{O}_ X)$-modules. For example $f$ gives rise to a morphism of ringed spaces $f' : (X, \mathcal{O}_ X) \to (Y, f_*\mathcal{O}_ X)$. By Lemma 20.13.3 the terms $E_ r^{p, q}$ of the Leray spectral sequence for an $\mathcal{O}_ X$-module $\mathcal{F}$ and $f$ are identical with those for $\mathcal{F}$ and $f'$ at least for $r \geq 2$. Namely, they both agree with the terms of the Leray spectral sequence for $\mathcal{F}$ as an abelian sheaf. And since $(f_*\mathcal{O}_ X)(Y) = \mathcal{O}_ X(X)$ we see the result. It is often the case that the Leray spectral sequence carries additional structure.
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)