Spectral sequence

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 $$r \ge r_0$$:

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.