$\cdots \to H^{p+n} \rightarrow E_{2}^{p, n} \to E_{2}^{p+n+1,0} \to H^{p+n+1} \to E_{2}^{p+1, n} \to E_{2}^{p+n+2,0} \to \cdots$

Question

Suppose $E_{2}^{p q}=0$ unless $q=0$ or $q=n$, for some $n \geq 2$. How to show that there is a long exact sequence $$ \cdots \rightarrow H^{p+n} \rightarrow E_{2}^{p, n} \rightarrow E_{2}^{p+n+1,0} \rightarrow H^{p+n+1} \rightarrow E_{2}^{p+1, n} \rightarrow E_{2}^{p+n+2,0} \rightarrow \cdots $$

Could you please give me any hints? Thanks.

Answer 1

Let's write the differentials as $d_{r}^{p,q} : E_{r}^{p,q} \to E_{r}^{p+r,q-r+1}$. Then the map $E_{i+1}^{p,q} \to E_{i}^{p,q}$ is an equality for $i \ne n$ since $d_{i}^{p,q} = 0$ for $i \ne n+1$; in particular, the spectral sequence converges on the $(n+2)$th page. Thus there is an exact sequence \begin{align} 0 \to \operatorname{coker} d_{n+1}^{p-1,n} \to H^{p+n} \to \ker d_{n+1}^{p,n} \to 0 \end{align} for all $p$.