Spectral sequence

Definition

A spectral sequence in an abelian category $C$ is a bunch of data of the following form:

A choice of integer $r_{0}$
For every integer $r \geq r_{0}$ , an object $E_{r}$ of $C$ , called a sheet or page or term.
For every integer $r \geq r_{0}$ , a morphism $d_{r} : E_{r} \to E_{r}$ called a boundary map or differential such that $d_{r} \circ d_{r} = 0$ , where $0$ is the zero map for the abelian category $C$ .
For every integer $r \geq r_{0}$ , an isomorphism between the homology object $H (E_{r})$ and $E_{r + 1}$ . Note that homology object here means the quotient $K e r (E_{r}) / I m (E_{r})$ . This makes sense because $C$ is an abelian category.

Suppose $C$ is a chain complex of abelian groups (or more generally, modules over a fixed commutative unital ring). Consider the category of chain complexes with chain maps. $C$ is an object in this category. We consider a natural spectral sequence in the same category associated with $C$ .