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snake lemma (Theorem)

Let $ \mathcal{A}$ be an abelian category. The snake lemma consists of the following two claims:

  1. Suppose
    $\displaystyle \begin{CD} 0@>>> A_1@>>> B_1@>>> C_1@>>> 0\ & & @V\alpha VV @V\beta VV @V\gamma VV\ 0@>>> A_2@>>>B_2@>>>C_2@>>>0 \end{CD}$
    is a commutative diagram in $ \mathcal{A}$ with exact rows. Then there is an exact sequence
    $\displaystyle 0 \to \mathrm{ker}\,\alpha \to \mathrm{ker}\,\beta \to \mathrm{ke... ...thrm{coker}\,\alpha \to \mathrm{coker}\,\beta \to \mathrm{coker}\,\gamma\to 0, $
    usually called the kernel-cokernel sequence. The morphism $ s$ is called the connecting morphism.
  2. Applying the previous claim inductively, for any short exact sequence
    $\displaystyle 0 \to \mathbf{A} \to \mathbf{B} \to \mathbf{C} \to 0 $
    of chain complexes in $ \mathcal{A}$, there is a corresponding long exact sequence in homology
    $\displaystyle \cdots \to H_n(\mathbf{A})\to H_n(\mathbf{B})\to H_n(\mathbf{C}) \to H_{n-1}(\mathbf A)\to\cdots $



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"snake lemma" is owned by mps. [ full author list (2) | owner history (1) ]
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Other names:  zig-zag lemma, serpent lemma

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proof of snake lemma (Proof) by mps
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Cross-references: chain complexes, short exact sequence, morphism, exact sequence, commutative diagram, abelian category
There are 4 references to this entry.

This is version 9 of snake lemma, born on 2002-12-13, modified 2006-02-15.
Object id is 3745, canonical name is SnakeLemma.
Accessed 5817 times total.

Classification:
AMS MSC: 18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes)
Pending Errata and Addenda
None.
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