The Stacks project

Definition 12.20.1. Let $\mathcal{A}$ be an abelian category.

1. A spectral sequence in $\mathcal{A}$ is given by a system $(E_ r, d_ r)_{r \geq 1}$ where each $E_ r$ is an object of $\mathcal{A}$, each $d_ r : E_ r \to E_ r$ is a morphism such that $d_ r \circ d_ r = 0$ and $E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(d_ r)$ for $r \geq 1$.

2. A morphism of spectral sequences $f : (E_ r, d_ r)_{r \geq 1} \to (E'_ r, d'_ r)_{r \geq 1}$ is given by a family of morphisms $f_ r : E_ r \to E'_ r$ such that $f_ r \circ d_ r = d'_ r \circ f_ r$ and such that $f_{r + 1}$ is the morphism induced by $f_ r$ via the identifications $E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(d_ r)$ and $E'_{r + 1} = \mathop{\mathrm{Ker}}(d'_ r)/\mathop{\mathrm{Im}}(d'_ r)$.