Spectral sequence

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

Definition

A (cohomology) spectral sequence (starting at $E_{a}$) in an abelian category $A$ consists of the following data:

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

$E_{r+1}^{pq}\simeq \ker(d_{r}^{pq})/{\text{image}}(d_{r}^{p-r,q+r+1})$

Examples

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