Snake Lemma

The Snake Lemma is one of the most important tools in homological algebra. It produces a long exact sequence from a morphism of short exact sequences, providing the connecting homomorphism that is the engine behind the long exact sequence in cohomology.

Statement

Theorem3.6Snake Lemma

In an abelian category, given a commutative diagram with exact rows:

\begin{array}{ccccccccc} & & A' & \xrightarrow{f'} & B' & \xrightarrow{g'} & C' & \to & 0 \\ & & \downarrow\scriptstyle{\alpha} & & \downarrow\scriptstyle{\beta} & & \downarrow\scriptstyle{\gamma} & & \\ 0 & \to & A & \xrightarrow{f} & B & \xrightarrow{g} & C & & \end{array}

there exists an exact sequence:

$\ker \alpha \xrightarrow{\bar{f'}} \ker \beta \xrightarrow{\bar{g'}} \ker \gamma \xrightarrow{\delta} \mathrm{coker}\, \alpha \xrightarrow{\bar{f}} \mathrm{coker}\, \beta \xrightarrow{\bar{g}} \mathrm{coker}\, \gamma$

Moreover, if $f'$ is monic (top row is a SES), then $\bar{f'}$ is monic. If $g$ is epic (bottom row is a SES), then $\bar{g}$ is epic. In the case of a morphism of short exact sequences, we get:

$0 \to \ker \alpha \to \ker \beta \to \ker \gamma \xrightarrow{\delta} \mathrm{coker}\, \alpha \to \mathrm{coker}\, \beta \to \mathrm{coker}\, \gamma \to 0$

The Connecting Homomorphism

RemarkConstruction of delta

The connecting homomorphism $\delta : \ker \gamma \to \mathrm{coker}\, \alpha$ is constructed by "chasing the diagram": given $c' \in \ker \gamma$ , lift to $b' \in B'$ (using surjectivity of $g'$ ), map to $\beta(b') \in B$ , observe that $g(\beta(b')) = \gamma(g'(b')) = 0$ , so $\beta(b') \in \ker g = \mathrm{im}\, f$ , lift to $a \in A$ , and project to $\mathrm{coker}\, \alpha$ . Independence of choices follows from exactness.

Examples

ExampleSnake Lemma in R-Mod

For $R$ -modules and a commutative diagram with exact rows, the Snake Lemma gives an explicit long exact sequence. The connecting map $\delta$ traces elements through the diagram: start in $\ker \gamma$ , go up, left, down, and project.

ExampleLong exact sequence from a SES of complexes

The long exact sequence in cohomology $\cdots \to H^n(A) \to H^n(B) \to H^n(C) \xrightarrow{\delta} H^{n+1}(A) \to \cdots$ is derived by applying the Snake Lemma to the diagram of cokernels and kernels of the differentials. This is proved in detail in the proof via Snake Lemma.

ExampleMultiplication by n on abelian groups

For the map $\times n : A \to A$ on an abelian group, applied to a SES $0 \to A \to B \to C \to 0$ , the Snake Lemma gives:

$0 \to A[n] \to B[n] \to C[n] \xrightarrow{\delta} A/nA \to B/nB \to C/nC \to 0$

where $A[n] = \ker(\times n)$ is the $n$ -torsion.

ExampleSnake Lemma and Tor

The long exact sequence for $\mathrm{Tor}$ arises from applying the Snake Lemma (or its derived version) to the tensor product of a flat resolution with a short exact sequence.

ExampleCohomology of mapping cone

The Snake Lemma applied to the mapping cone triangle gives the long exact sequence $\cdots \to H^n(A) \to H^n(B) \to H^n(\mathrm{Cone}(f)) \to H^{n+1}(A) \to \cdots$ .

ExampleKummer sequence in algebraic geometry

The Kummer sequence $0 \to \mu_n \to \mathbb{G}_m \xrightarrow{n} \mathbb{G}_m \to 0$ on an algebraic variety gives, via the long exact sequence in etale cohomology:

$\cdots \to H^1_{\mathrm{et}}(X, \mu_n) \to \mathrm{Pic}(X) \xrightarrow{n} \mathrm{Pic}(X) \to H^2_{\mathrm{et}}(X, \mu_n) \to \cdots$

ExampleSnake Lemma for exact couples

The Snake Lemma is used in the construction of spectral sequences from exact couples: the connecting homomorphism provides the differential of the derived exact couple.

ExampleSnake Lemma in abelian category via Freyd-Mitchell

In a general abelian category (not necessarily a module category), the Snake Lemma holds because the Freyd-Mitchell embedding allows us to reduce to $R\text{-}\mathbf{Mod}$ and chase elements.

ExampleSnake Lemma and the 9-lemma

The 9-lemma (or $3 \times 3$ lemma) states that in a $3 \times 3$ commutative diagram with exact columns, if two rows are exact, so is the third. This follows from two applications of the Snake Lemma.

ExampleSnake Lemma applied to localization

For a ring $R$ , prime $\mathfrak{p}$ , and a SES $0 \to M' \to M \to M'' \to 0$ , localization at $\mathfrak{p}$ gives $0 \to M'_\mathfrak{p} \to M_\mathfrak{p} \to M''_\mathfrak{p} \to 0$ . The Snake Lemma (applied to the localization maps) relates the kernels and cokernels of localization.

ExampleConnecting homomorphism in sheaf cohomology

For a SES of sheaves $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ on $X$ , the global sections functor gives $0 \to \Gamma(\mathcal{F}') \to \Gamma(\mathcal{F}) \to \Gamma(\mathcal{F}'')$ , and the connecting homomorphism $\delta : \Gamma(\mathcal{F}'') \to H^1(X, \mathcal{F}')$ measures the obstruction to lifting global sections.

ExampleSnake Lemma and Euler characteristic

For a SES $0 \to A \to B \to C \to 0$ of finite-length modules, the Snake Lemma implies $\ell(B) = \ell(A) + \ell(C)$ , where $\ell$ denotes length. More generally, exact sequences are additive with respect to Euler characteristic.

Significance

RemarkThe engine of homological algebra

The Snake Lemma is used to:

Construct the long exact sequence in cohomology.
Prove the Five Lemma.
Build long exact sequences for derived functors ( $\mathrm{Ext}$ , $\mathrm{Tor}$ , sheaf cohomology).
Establish connecting homomorphisms in spectral sequences.