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  E a {\displaystyle E_{a}} ) in an abelian category  A {\displaystyle A} consists of the following data:

1. A family  { E r p q } {\displaystyle \\{E_{r}^{pq}\\}} of objects of  A {\displaystyle A} defined for all integers  p , q {\displaystyle p,q} and  r ≥ a {\displaystyle r\geq a}
2. Morphisms  d r p q : E r p q -> E r p + r , q − r + 1 {\displaystyle d_{r}^{pq}:E_{r}^{pq}\to E_{r}^{p+r,q-r+1}} that are differentials in the sense that  d r ∘ d r = 0 {\displaystyle d_{r}\circ d_{r}=0} , so that the lines of "slope"  − r / ( r + 1 ) {\displaystyle -r/(r+1)} in the lattice  E r ∗ ∗ {\displaystyle E_{r}^{**}} form chain complexes (we say the differentials "go to the right")
3. Isomorphisms between  E r + 1 p q {\displaystyle E_{r+1}^{pq}} and the homology of  E r ∗ ∗ {\displaystyle E_{r}^{**}} at the spot  E r p q {\displaystyle E_{r}^{pq}} :

E r + 1 p q ≃ ker ⁡ ( d r p q ) / image ( d r p − r , q + r + 1 ) {\displaystyle E_{r+1}^{pq}\simeq \ker(d_{r}^{pq})/{\text{image}}(d_{r}^{p-r,q+r+1})}

## Convergence[edit]

## Examples[edit]

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