Lemma 10.4.1. Given a commutative diagram

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

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.

[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.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: