Long exact sequence in homology

Remarks

For modules, $\partial_n$ is computed as follows. Lift a cycle $z \in C_n$ to $y \in B_n$. Then $d y$ maps to $d z = 0$ in $C_{n-1}$, so $d y = f(x)$ for a unique $x \in A_{n-1}$, which is a cycle, and $\partial_n [z] = [x]$. The proof applies the snake lemma to the map $\operatorname{coker} d^A_{n+1} \to \ker d^A_{n-1}$ induced by $d$, and the same for $B$ and $C$. The kernels and cokernels of these maps are $H_n$ and $H_{n-1}$, which yields the six-term pieces of the sequence. Example: let $A_\bullet = \Z$ in degree $0$, let $B_\bullet$ have $B_1 = B_0 = \Z$ with $d_1 = 1_{\Z}$, and let $C_\bullet = \Z$ in degree $1$, with the evident inclusion and projection. Then $H(B) = 0$, so exactness of $0 = H_1(B) \to H_1(C) \to H_0(A) \to H_0(B) = 0$ makes $\partial_1 \colon \Z \to \Z$ an isomorphism.

Going deeper

For a student

This is the most used tool for computing homology. In topology, the short exact sequences of singular chains for a pair $(X, A)$ give the long exact sequence of a pair. For an open cover by two sets they give the Mayer–Vietoris sequence. In algebra, applying it to a short exact sequence of resolutions gives the long exact sequences of derived functors such as $\mathrm{Ext}$ and $\mathrm{Tor}$. Together with the five lemma, it lets one prove that a map is a quasi-isomorphism by checking pieces.