Spectral sequence

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

Spectral sequences were invented by Jean Leray as an approach to computing sheaf cohomology.

## Contents

* 1 Historical development
* 2 Definition
* 3 Convergence
* 4 Examples

## Historical development[edit]

## Definition[edit]

A (cohomology) spectral sequence (starting at Ea) in an abelian category A consists of the following data:

1. A family {Erpq} of objects of A defined for all integers p,q and r≥a
2. Morphisms drpq:Erpq->Erp+r,q−r+1 that are differentials in the sense that dr∘dr=0, so that the lines of "slope" −r/(r+1) in the lattice Er** form chain complexes (we say the differentials "go to the right")
3. Isomorphisms between Er+1pq and the homology of Er** at the spot Erpq:

Er+1pq≃ker⁡(drpq)/image(drp−r,q+r+1)

## Convergence[edit]

## Examples[edit]

1. The Leray spectral sequence
2. The Grothendieck spectral sequence