# nLab Leray spectral sequence

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

### Theorems

#### Topos Theory

Could not include topos theory - contents

cohomology

# Contents

## 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

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

that converges as

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

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 $A$.

## References

Lecture notes include

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

Textbook accounts with an eye specifically towards étale cohomology

Revised on November 26, 2013 01:35:09 by Urs Schreiber (145.116.131.216)