Spectral sequence

Definition

Definition of spectral sequence in an abelian category

A spectral sequence in an abelian category ${\mathcal {C}}$ is defined as the following data: a nonnegative integer $r_{0}$ along with the following for each integer $r\geq r_{0}$:

Notation	Type	Condition	Name
$E_{r}$	object in ${\mathcal {C}}$	--	sheet, page, or term
$d_{r}:E_{r}\to E_{r}$	endomorphism of $E_{r}$ with respect to category ${\mathcal {C}}$	$d_{r}\circ d_{r}=0$	boundary map or differential
isomorphism $E_{r+1}\cong H(E_{r})$	isomorphism is in ${\mathcal {C}}$, $H$ denotes the homology object		We say that $E_{r+1}$ is the derived object' of $E_{r}$

Definition of spectral sequence in the category of chain complexes of abelian groups

Suppose ${\mathcal {C}}$ is the category of chain complexes of abelian groups. Consider a spectral sequence with $r_{0}=0$. Then, $E_{0}$ could be any chain complex with any differential. $E_{1}\cong H(E_{0})$ will then be a chain complex with zero differential. We will then have $E_{2}\cong H(E_{1})\cong E_{1}$, and in fact, all the higher sheets will be isomorphic to $E_{1}$. In other words, the spectral sequence stabilizes at $E_{1}$.

Definition of spectral sequence in the category of doubly graded modules over a commutative unital ring

Consider the category of doubly graded modules over a commutative unital ring. Each sheet here is a doubly graded module:

$E_{r}=\bigoplus _{p,q}E_{r}^{p,q}$

And the differential $d_{r}$ has bidegree $(-r,r-1)$, i.e., $d_{r}$ defines a bunch of morphisms:

$E_{r}^{p,q}\to E_{r}^{p-r,q+r-1}$

for each $p,q$, subject to the composite of two composable morphisms of this sort being zero.

• For $r=0$, all the differentials are vertically downward.
• For $r=1$, all the differentials are horizontally rightward.
• For $r=2$, the differentials are in "knight's move" directions.