Snake lemma

Description

The snake lemma is a tool from homological algebra, used to calculate the connecting homomorphism in a Long exact sequence.

Definition

Let the above be a commutative diagram of Abelian groups where both lines are exact sequences. Then the snake lemma tells us that there is a Long exact sequence $\text{ker }a\to \text{ker }b\to \text{ker }c\stackrel{d}{\to }\text{coker }a\to \text{coker }b\to \text{coker }c$ where $d$ is the connected homomorphism often encountered in algebraic topology.

Proof: We will only prove the existence of the connecting homomorphism. Let $\gamma$ be an element of $\text{ker }c$. Because $g$ is surjective, there is a $\beta$ s.t. $g(\beta )=\gamma$. Pushing down along $b$ gives us an element ${\beta }^{\prime }$. Because $\gamma$ lies in $\text{ker }c$, by commutativity, we have ${\beta }^{\prime }\in \text{ker }{g}^{\prime }$ so by exactness we can find an element ${\alpha }^{\prime }$ representing this element. This will then be mapped down to ${\alpha }^{\prime }+\text{im }a$ giving us the cokernel. However, it is not clear whether this is natural. We do the proof, that it is natural. Take a different choice $\bar{\beta }$ s.t. $g(\bar{\beta })=\gamma$. Then $g(\beta -\bar{\beta })=0$ and therefore $\beta -\bar{\beta }\in \text{ker }g$. Exactness tells us that there is an $\alpha$ with $f(\alpha )=\beta -\bar{\beta }$. We push down along $b$ and pull back along $f^{\prime }$ to get ${\alpha }^{\prime }-{\bar{\alpha }}^{\prime }$. Due to the injectivity of $f^{\prime }$ and the commutativity of the diagram this must be the image of $\alpha$ under $a$. It follows that ${\alpha }^{\prime }-{\bar{\alpha }}^{\prime }$ is part of the image of $a$ and trivial in the cokernel. In different words, ${\alpha }^{\prime }$ and ${\bar{\alpha }}^{\prime }$ map to the same element in the cokernel.

The proof of the snake lemma is important. The snake lemma tells us that the connecting map exists and that the construction is natural but we need the proof to actually do the construction.

Properties

Tip

Examples

Example