# nLab snake lemma

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

A basic lemma in homological algebra: it constructs connecting homomorphisms.

## Statement

###### Lemma

Let

$\begin{array}{ccccccccc}& & A\prime & \to & B\prime & \stackrel{p}{\to }& C\prime & \to & 0\\ & & {↓}^{f}& & {↓}^{g}& & {↓}^{h}\\ 0& \to & A& \stackrel{i}{\to }& B& \to & C\end{array}$\array{ && A' &\to & B' &\stackrel{p}{\to}& C' &\to & 0 \\ && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{h}} \\ 0 &\to& A &\stackrel{i}{\to} & B &\to& C }

be a commuting diagram in an abelian category $𝒜$ such that the two rows are exact sequences.

Then there is a long exact sequence of kernels and cokernels of the form

$\mathrm{ker}\left(f\right)\to \mathrm{ker}\left(g\right)\to \mathrm{ker}\left(h\right)\stackrel{\partial }{\to }\mathrm{coker}\left(f\right)\to \mathrm{coker}\left(g\right)\to \mathrm{coker}\left(h\right)\phantom{\rule{thinmathspace}{0ex}}.$ker(f) \to ker(g) \to ker(h) \stackrel{\partial}{\to} coker(f) \to coker(g) \to coker(h) \,.

Moreover

• if $A\to B$ is a monomorphism then so is $\mathrm{ker}\left(f\right)\to \mathrm{ker}\left(g\right)$

• if $B\to C$ is an epimorphism, then so is $\mathrm{coker}\left(g\right)\to \mathrm{coker}\left(h\right)$.

If $𝒜$ is realized as a (full subcategory of) a category of $R$-modules, then the connecting homomorphism $\partial$ here can be defined on elements $c\prime \in \mathrm{ker}\left(h\right)\subset C\prime$ by

$\partial \left(c\prime \right):={i}^{-1}\phantom{\rule{thinmathspace}{0ex}}g\phantom{\rule{thinmathspace}{0ex}}{p}^{-1}\left(c\prime \right)\phantom{\rule{thinmathspace}{0ex}},$\partial (c') := i^{-1} \,g\, p^{-1}(c') \,,

where ${i}^{-1}\left(-\right)$ and ${p}^{-1}\left(-\right)$ denote any choice of pre-image (the total formula is independent of that choice).

###### Remark

The snake lemma derives its name from the fact that one may draw the connecting homomorphism $\partial$ that it constructs diagrammatically as follows:

## References

An early occurence of the snake lemma is as lemma (5.8) of

• D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (JSTOR)

In

it appears as lemma 1.3.2.

A purely category-theoretic proof is given in

• Temple Fay, Keith Hardie, Peter Hilton, The two-square lemma, Publicacions Matemàtiques, Vol 33 (1989) (pdf)

and in