# nLab Leray spectral sequence

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# 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, 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 $𝒞={\mathrm{Ch}}_{•}\left(\mathrm{Sh}\left(X,\mathrm{Ab}\right)\right)$ and $𝒟={\mathrm{Ch}}_{buller}\left(\mathrm{Sh}\left(Y,\mathrm{Ab}\right)\right)$ be the model categories of complexes of sheaves of abelian groups. The direct image ${f}_{*}$ and global section functor ${\Gamma }_{Y}$ compose to ${\Gamma }_{X}$:

${\Gamma }_{X}:𝒞\stackrel{{f}_{*}}{\to }𝒟\stackrel{{\Gamma }_{Y}}{\to }{\mathrm{Ch}}_{•}\left(\mathrm{Ab}\right)\phantom{\rule{thinmathspace}{0ex}}.$\Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab) \,.

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

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

that converges as

${E}_{r}^{p,q}⇒{H}^{p+q}\left(X,A\right)$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 $A$.

Revised on October 29, 2012 13:09:58 by Urs Schreiber (89.204.137.136)