## You are here

Homesnake lemma

## Primary tabs

# 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

## Attached Articles

## Corrections

classification by akrowne ✓

this is a Theorem by paolini ✓

synonyms by mps ✓

typo and capitalisation in synonyms by mps ✓

Ambiguous by mathcam ✓

this is a Theorem by paolini ✓

synonyms by mps ✓

typo and capitalisation in synonyms by mps ✓

Ambiguous by mathcam ✓