Spectral sequence

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 $$r \ge r_0$$, an object $$E_r$$ of $$\mathbb{C}$$, called a sheet or page or term.
 * For every integer $$r \ge r_0$$, 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 $$r \ge r_0$$, 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 $$r \ge 1$$, define $$E_r = H(C)$$ and $$d_r$$ to be the zero map.