## Tag `07JV`

## 10.4. Snake lemma

The snake lemma and its variants are discussed in the setting of abelian categories in Homology, Section 12.5.

Lemma 10.4.1. Suppose 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, then 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, and 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. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 287–334 (see updates for more information).

```
\section{Snake lemma}
\label{section-snake}
\noindent
The snake lemma and its variants are discussed in the setting of
abelian categories in
Homology, Section \ref{homology-section-abelian-categories}.
\begin{lemma}
\label{lemma-snake}
\begin{reference}
\cite[III, Lemma 3.3]{Cartan-Eilenberg}
\end{reference}
Suppose 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, then there is a canonical exact sequence
$$
\Ker(\alpha) \to \Ker(\beta) \to \Ker(\gamma)
\to
\Coker(\alpha) \to \Coker(\beta) \to \Coker(\gamma)
$$
Moreover, if $X \to Y$ is injective, then the first map is
injective, and if $V \to W$ is surjective, then the last
map is surjective.
\end{lemma}
\begin{proof}
The map $\partial : \Ker(\gamma) \to \Coker(\alpha)$ is defined
as follows. Take $z \in \Ker(\gamma)$. 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.
\end{proof}
```

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