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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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