[III, Lemma 3.3, Cartan-Eilenberg]

Lemma 10.4.1. Given a commutative diagram

$\xymatrix{ & X \ar[r] \ar[d]^\alpha & Y \ar[r] \ar[d]^\beta & Z \ar[r] \ar[d]^\gamma & 0 \\ 0 \ar[r] & U \ar[r] & V \ar[r] & W }$

of abelian groups with exact rows, there is a canonical exact sequence

$\mathop{\mathrm{Ker}}(\alpha ) \to \mathop{\mathrm{Ker}}(\beta ) \to \mathop{\mathrm{Ker}}(\gamma ) \to \mathop{\mathrm{Coker}}(\alpha ) \to \mathop{\mathrm{Coker}}(\beta ) \to \mathop{\mathrm{Coker}}(\gamma )$

Moreover: if $X \to Y$ is injective, then the first map is injective; if $V \to W$ is surjective, then the last map is surjective.

Proof. The map $\partial : \mathop{\mathrm{Ker}}(\gamma ) \to \mathop{\mathrm{Coker}}(\alpha )$ is defined as follows. Take $z \in \mathop{\mathrm{Ker}}(\gamma )$. Choose $y \in Y$ mapping to $z$. Then $\beta (y) \in V$ maps to zero in $W$. Hence $\beta (y)$ is the image of some $u \in U$. Set $\partial z = \overline{u}$, the class of $u$ in the cokernel of $\alpha$. Proof of exactness is omitted. $\square$

There are also:

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