The Stacks project

[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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 10.4: Snake lemma

Post a comment

Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07JW. Beware of the difference between the letter 'O' and the digit '0'.


View Lemma 10.4.1 as pdf