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Homesnake lemma

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# snake lemma

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

1. Suppose

$\begin{matrix}0&\cd@stack{\rightarrowfill@}{}{}&A_{1}&\cd@stack{% \rightarrowfill@}{}{}&B_{1}&\cd@stack{\rightarrowfill@}{}{}&C_{1}&\cd@stack{% \rightarrowfill@}{}{}&0\\ &&{\alpha}{\Big\downarrow}&&{\beta}{\Big\downarrow}&&{\gamma}{\Big\downarrow}&% &\\ 0&\cd@stack{\rightarrowfill@}{}{}&A_{2}&\cd@stack{\rightarrowfill@}{}{}&B_{2}&% \cd@stack{\rightarrowfill@}{}{}&C_{2}&\cd@stack{\rightarrowfill@}{}{}&0\end{matrix}$ is a commutative diagram in $\mathcal{A}$ with exact rows. Then there is an exact sequence

$0\to\mathrm{ker}\,\alpha\to\mathrm{ker}\,\beta\to\mathrm{ker}\,\gamma\stackrel% {s}{\longrightarrow}\mathrm{coker}\,\alpha\to\mathrm{coker}\,\beta\to\mathrm{% coker}\,\gamma\to 0,$ 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

$0\to\mathbf{A}\to\mathbf{B}\to\mathbf{C}\to 0$ of chain complexes in $\mathcal{A}$, there is a corresponding long exact sequence in homology

$\cdots\to H_{n}(\mathbf{A})\to H_{n}(\mathbf{B})\to H_{n}(\mathbf{C})\to H_{{n% -1}}(\mathbf{A})\to\cdots$

Synonym:

zig-zag lemma, serpent lemma

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

18G35*no label found*

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