An abelian group $A$ with a homomorphism $d: A \longrightarrow A$ is called differential group if $d^2 = 0$.
Let $(A, d)$ be a differential group, $B \leq A$, $d: B \longrightarrow B$. Then $A, B, \frac{A}{B}$ are differential groups, I'll use the same $d$ for the differentials.
We may take a look on the short exact sequence:
$0 \to B \stackrel{i}{\to} A \stackrel{j}{\to} \frac{A}{B} \to 0$.
Now to the point of my question, I read that taking the homology of this exact sequence gives us an exact sequnce which is called the homology exat sequnce of pair of differential groups:
$... \stackrel{j_\star}{\to} H(\frac{A}{B}, d) \stackrel{\partial}{\to} H(B, d) \stackrel{i_\star}{\to} H(A, d) \stackrel{j_\star}{\to} H(\frac{A}{B}, d) \stackrel{\partial}{\to} H(B, d) \stackrel{i_\star}{\to} ...$.
I am cheking the exatness. I cheked the exactness in $H(A, d), H(B, d)$ but I get nothing good in the $H(\frac{A}{B}, d)$. Can you help me with that?
Ivo Terek gave the answer in the comments