nLab Leray spectral sequence

Contents

Context

Homological algebra

Topos Theory

Cohomology

Definition
Properties
Related concepts
References

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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Last revised on May 6, 2015 at 19:25:57. See the history of this page for a list of all contributions to it.

nLab Leray spectral sequence

Context

Homological algebra

Topos Theory

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In higher category theory

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