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Leray spectral sequence

Definition

The Leray spectral sequence is the special case of the Grothendieck spectral sequence for the case where the two functors being composed are a push-forward of sheaves of abelian groups along a continuous map $f:X\to Y$ between topological spaces, followed by the push-forward $Y\to *$ to the point – the global section functor. This yields a spectral sequence that computes the abelian sheaf cohomology on $X$ in terms of the abelian sheaf cohomology on $Y$.

Properties

Theorem

Let $X,Y$ be suitable sites let and $f:X\to Y$ be a morphism of sites. Let $𝒞={\mathrm{Ch}}_{•}(\mathrm{Sh}(X,\mathrm{Ab}))$ and $𝒟={\mathrm{Ch}}_{buller}(\mathrm{Sh}(Y,\mathrm{Ab}))$ be the model categories of complexes of sheaves of abelian groups. The direct image $f_{*}$ and global section functor ${\Gamma }_Y$ compose to ${\Gamma }_X$:

$${\Gamma }_X:𝒞\stackrel{f_{*}}{\to }𝒟\stackrel{{\Gamma }_Y}{\to }{\mathrm{Ch}}_{•}(\mathrm{Ab}).$$\Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab) \,.

Then for $A\in \mathrm{Sh}(X,\mathrm{Ab})$ a sheaf of abelian groups on $X$ there is a cohomology spectral sequence

$$E_r^{p,q}:=H^p(Y,R^q{f}_{*}A)$$E_r^{p,q} := H^p(Y, R^q f_* A)

that converges as

$$E_r^{p,q}\Rightarrow H^{p+q}(X,A)$$E_r^{p,q} \Rightarrow H^{p+q}(X, A)

and hence computes the cohomology of $X$ with coefficients in $A$ in terms of the cohomology of $Y$ with coefficients in the push-forward of $A$.

Related concepts

• Leray-Serre spectral sequence