nLab 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 or more generally the direct image of a morphism of sites, 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 $\mathcal{C} = Ch_\bullet(Sh(X,Ab))$ and $\mathcal{D} = Ch_\bullet(Sh(Y,Ab))$ be the model categories of complexes of sheaves of abelian groups. The direct image $f_*$ and global section functor $\Gamma_Y$ compose to $\Gamma_X$:

$$\Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab) \,.$$

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

$$E_2^{p,q} := H^p(Y, R^q f_* A)$$

that converges as

$$E_2^{p,q} \Rightarrow H^{p+q}(X, A)$$

and hence computes the abelian sheaf cohomology of $X$ with coefficients in $A$ in terms of the cohomology of $Y$ with coefficients in the derived direct image of $A$.

Related concepts

• Leray-Serre spectral sequence

References

Lecture notes include

• Dan Petersen, Leray spectral sequence, November 2010 (pdf)

• Greg Friedman, Some extremely brief notes on the Leray spectral sequence (pdf)

Textbook accounts with an eye specifically towards étale cohomology

• Günter Tamme, section I 3.7 of Introduction to Étale Cohomology

• James Milne, section 11 of Lectures on Étale Cohomology

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