TITLE: Weibel HA Exercise 1.5.9
QUESTION [12 upvotes]: I cannot solve the following problem from Weibel:

Let $f:B\to C$ be a map of chain complexes. Show that the natural maps $\alpha : \ker(f)[-1]\to \operatorname{cone}(f)$ and $\beta: \operatorname{cone}(f)\to \operatorname{coker}(f)$ give rise to a long exact sequence:
  $$\cdots \to H_{n-1}(\ker(f))\to H_n(\operatorname{cone}(f))\to H_n(\operatorname{coker}(f))\to H_{n-2}(\ker(f)) \to \cdots$$

Any help is appreciated!

REPLY [8 votes]: Here's a sketch. The details it leaves to be checked are fairly straightforward, I hope.
Let $g:\text{im}(f)\to C$ be the inclusion map.
There is a short exact sequence of complexes
$$0\to\ker(f)[-1]\to\text{cone}(f)\to\text{cone}(g)\to0,$$
which gives a long exact sequence of homology.
The natural map $\text{cone}(f)\to\text{cok}(f)$ factors through $\text{cone}(g)$, with the resulting map $\text{cone}(g)\to\text{cok}(f)$ a quasi-isomorphism.