## How the Staircase Diagram changes when we pass to derived couple (Spectral Sequence)

Set $A_{n,p}^1=H_n(X_p)$ and $E_{n,p}^1=H_n(X_p,X_{p-1})$. The diagram then has the following form:

When we pass to the derived couple, each group $A_{n,p}^1$ is replaced by a subgroup $A_{n,p}^2=\text{Im}\,(i_1: A_{n,p-1}^1\to A_{n,p}^1)$. The differentials $d_1=j_1k_1$ go two units to the right, and we replace the term $E_{n,p}^1$ by the term $E_{n,p}^2=\text{Ker}\, d_1/\text{Im}\,d_1$, where the $d_1$‘s refer to the $d_1$‘s leaving and entering $E_{n,p}^1$ respectively.

The maps $j_2$ now go diagonally upward because of the formula $j_2(i_1a)=[j_1a]$. The maps $i_2$ and $k_2$ still go vertically and horizontally, $i_2$ being a restriction of $i_1$ and $k_2$ being induced by $k_1$.