Short Exact Sequence of Complexes Induces Long Exact Sequence of Homology Groups

abelian-categorieshomological-algebrahomology-cohomology

I am following Lang's Algebra on General Homology Theory and wanted to try proving the short exact sequence of complexes
$$\require{AMScd}
\begin{CD}
0 @>{}>> A @>{f}>> B @>{g}>> C @>{}>> 0
\end{CD}$$
induces a long exact sequence on the homology groups
$$\require{AMScd}
\begin{CD}
@>{}>> H^{i}(A) @>{}>> H^i(B) @>{}>> H^i(C) @>{\delta}>> H^{i+1}(A) @>{}>> H^{i+1}(B) @>{}>> H^{i+1}(C) @>{}>>
\end{CD}$$
by the snake lemma.
I saw this question regarding the same topic, and I am trying to flesh out the proof by following the comment noting the following diagram commutes and is exact in the rows.
$$\require{AMScd}
\begin{CD}
&&A_n/B(A_n) @>{f_{*}}>> B_n/B(B_n) @>{g_{*}}>> C_n/B(C_n) @>{}>> 0 \\
&@VVV @VVV @VVV \\
0 @>{}>> Z(A_{n+1}) @>{\alpha}>> Z(B_{n+1}) @>{\beta}>> Z(C_{n+1})
\end{CD}$$
I see how $f$ and $g$ induce homomorphisms $f_*, g_*$ by sending $f_*(a+B(A_n)) = f(a) + B(B_n)$ and similarly for $g$ (at least I think), but I cannot see how the top row is exact. I realize under the induced homomorphism $f_*$ we have $\text{Im}(f_*) = \text{Im}(f)/B(B_n)$ but I'm not sure how to prove this is $\ker(g_*) = g^{-1}(B(C_n))$.

Best Answer

There is probably a quick proof by abstract nonsense, but I can't see that argument right now. However, when in doubt, a diagram chase always works in entry-level homological algebra.

The more difficult direction is $\ker{g_*} \subseteq \operatorname{im}{f_*}$. Let $[b] \in \ker{g_*}$ with $b \in B_n$, so $g(b) \in B(C_n)$, i.e. there exists $c \in C_{n+1}$ such that $g(b) = d(c)$. By exactness of $0 \to A \to B \to C \to 0$ there exists $b' \in B_{n+1}$ such that $g(b') = c$. Because $g$ is a chain map, we have $$g(b) = d(g(b')) = g(d(b')) \implies g(b-d(b')) = 0.$$ Thus, $b-d(b') \in \ker{g}$, so by exactness there exists some $a \in A_n$ with $f(a) = b-d(b')$. But then $f([a]) = [b]$ which shows $[b] \in \operatorname{im}{f_*}$ as desired.

How did I find this proof? I really just followed my nose: I started taking the definition and writing down the only sensible thing one could do at each step. And just after a few lines we came to the desired conclusion!

In fact, it's probably easier to do this yourself than to read the proof of somebody else.

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