Spectral sequence of a filtered cochain complex

Definition

Let ${\mathcal {A}}$ be an abelian category. We describe below what a filtered cochain complex over ${\mathcal {A}}$ means and what it means to be a spectral sequence over such a filtered cochain complex.

Setup

• $C^{\cdot }$ is a cochain complex over ${\mathcal {A}}$. In other words, it has objects $C^{q},q\in \mathbb {Z}$, with morphisms $C^{q}\to C^{q+1}$ such that the composition of two successive morphisms of this sort gives the zero morphism.
• We are given a filtration of subcomplexes $F^{p}C^{\cdot }$ where $p\in \mathbb {Z}$. Each $F^{p}C^{\cdot }$ is a subcomplex of $C^{\cdot }$, and the filtration is descending, i.e., $F^{p}C^{\cdot }\supseteq F^{p+1}C^{\cdot }$.