nLab Leray spectral sequence

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Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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:XYf : X \to Y between topological spaces or more generally the direct image of a morphism of sites, followed by the push-forward Y*Y \to * to the point – the global section functor. This yields a spectral sequence that computes the abelian sheaf cohomology on XX in terms of the abelian sheaf cohomology on YY.

Properties

Theorem

Let X,YX, Y be suitable sites let and f:XYf : X \to Y be a morphism of sites. Let 𝒞=Ch (Sh(X,Ab))\mathcal{C} = Ch_\bullet(Sh(X,Ab)) and 𝒟=Ch (Sh(Y,Ab))\mathcal{D} = Ch_\bullet(Sh(Y,Ab)) be the model categories of complexes of sheaves of abelian groups. The direct image f *f_* and global section functor Γ Y\Gamma_Y compose to Γ X\Gamma_X:

Γ X:𝒞f *𝒟Γ YCh (Ab). \Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab) \,.

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

E 2 p,q:=H p(Y,R qf *A) E_2^{p,q} := H^p(Y, R^q f_* A)

that converges as

E 2 p,qH p+q(X,A) E_2^{p,q} \Rightarrow H^{p+q}(X, A)

and hence computes the abelian sheaf cohomology of XX with coefficients in AA in terms of the cohomology of YY with coefficients in the derived direct image of AA.

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

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