Spectral sequence

Definition

General definition

A spectral sequence in an abelian category $\mathbb {C}$ 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 $\mathbb {C}$, 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 $\mathbb {C}$.
• 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 $\operatorname {Ker} (E_{r})/\operatorname {Im} (E_{r})$. This makes sense because $\mathbb {C}$ is an abelian category.

Application to chain complex of abelian groups

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 $C$.

• Set $r_{0}=0$.
• Define $E_{0}=C$ and $d_{0}$ to be the differential of $C$.
• Define $E_{1}=H(C)$ and $d_{1}$ to be the zero map.
• For $r\geq 1$, define $E_{r}=H(C)$ and $d_{r}$ to be the zero map.