TITLE: Total complex homology exact sequence
QUESTION [8 upvotes]: I'm been trying to do this problem (Problem 5.1.1) from Weibel's Introduction to Homological Algebra but I can't really see how to finish it. The statement of the problem is summarized as follows:
Suppose that we have a double complex $E$ consists of only two columns $p$ and $p-1$. Let $T_n = Tot(E)$ be the total complex then show that there is an exact sequence
\begin{equation}
0 \rightarrow E^2_{p - 1,q + 1} \rightarrow H_{p + q} (T) \rightarrow E^2_{p,q} \rightarrow 0
\end{equation}
So what I've tried so far is attempting to calculate each of the object in the sequence and show that 
\begin{equation}
E^2_{p,q} \cong H_{p+q}(T)/E^2_{p-1,q+1}
\end{equation}
or something. So I did the calculation and I got
\begin{equation}
E^2_{p-1,q+1} \cong \mbox{ker} (d^v_{p-1, q+1})/\mbox{im}(d^h_{p, q+1}) \\
H_{p + q}(T) \cong \frac{(\mbox{ker}(d^h_{p, q}) \cap \mbox{ker}(d^v_{p,q})) \oplus \mbox{ker} (d^v_{p-1,q+1})}{\mbox{im} (d^v_{p,q+1}) \oplus (\mbox{im} (d^h_{p,q+1}) + \mbox{im} (d^v_{p-1,q+2}))} \\
E^2_{p,q} \cong \mbox{ker} ({d^h_{p,q}}_{\star})
\end{equation}
Where ${d^h_{p,q}}_{\star}: E^1_{p,q} \rightarrow E^1_{p-1,q}$ is the induced horizontal differential map after taking the vertical homology. However, 
\begin{equation}
H_{p+q}(T)/E^2_{p-1,q+1} \cong \frac{(\mbox{ker}(d^h_{p, q}) \cap \mbox{ker}(d^v_{p,q}))}{\mbox{im} (d^v_{p,q+1})} \stackrel{?}{\cong} \mbox{ker} ({d^h_{p,q}}_{\star})
\end{equation}
and I can't convince myself that the second equality is true.
Am I on the right track (or close to)? If someone could point out the mistake or guide me to the right direction that would be great. Thank you.

REPLY [5 votes]: The thing is in the first page $E^1_{*,*}$, the vertical map is actually zero. 
Now 
\begin{equation}
E^2_{p-1,q+1} \cong  \mbox{ker} (d^h_{p-1, q+1})/\mbox{im}(d^h_{p, q+1}) \cong E^1_{p-1,q+1}/\mbox{im}(d^h_{p, q+1}) \\
E^2_{p,q} \cong \mbox{ker} ({d^h_{p,q}})\\
H_{p + q}(T) \cong \frac{\{(a,b) | d^v_{p-1,q+1}(a)+d^h_{p,q}(b)=0;d^v_{p,q}(b)=0  \}}{\{(a,b) | a=d^v_{p-1,q+2}(x)+d^h_{p,q+1}(y); b=d^v_{p,q+1}(y);\}}=\frac{(E^1_{p-1,q+1},\mbox{ker} ({d^h_{p,q}}))}{(\mbox{im} (d^h_{p,q+1}),0)} \\
\end{equation}
Here $x \in E^1_{p-1,q+2}$ and $y \in E^1_{p,q+1}$.
Hence you get the desired exact sequence.
Note: your understanding of differentials in $\mbox{Tot}(E)$ is not right.