Snake lemma

Remarks

The morphism $\partial$ is called the connecting morphism. For modules, $\partial$ is defined by a chase. Given $z \in \ker c$, choose $y \in B$ with $g(y) = z$. Then $g'(b(y)) = c(z) = 0$, so $b(y) = f'(x')$ for a unique $x' \in A'$, and $\partial(z)$ is the class of $x'$ in $\operatorname{coker} a$. One checks that this class does not depend on the choice of $y$, and then checks exactness at each place by similar chases. In a general abelian category one uses pseudo-elements or an embedding theorem.

Example: map $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$ to itself by multiplication by $2$ in each place, which is $0$ on $\Z/2\Z$. The kernels are $0, 0, \Z/2\Z$ and the cokernels are $\Z/2\Z, \Z/2\Z, \Z/2\Z$. Exactness forces $\partial \colon \Z/2\Z \to \Z/2\Z$ to be an isomorphism, the map $\operatorname{coker} a \to \operatorname{coker} b$ to be $0$, and $\operatorname{coker} b \to \operatorname{coker} c$ to be an isomorphism, which one can confirm directly.

Going deeper

For a student

The snake lemma is the engine behind most long exact sequences. Applied to a short exact sequence of chain complexes, it gives the long exact sequence in homology. It also gives the long exact sequences of derived functors and the nine lemma. In practice, it is how one computes a kernel or cokernel that is hard to see directly, from ones that are easy to see.