The Stacks project

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.

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