diagram chasing in homological algebra
salamander lemma $\Rightarrow$
four lemma $\Rightarrow$ five lemma
snake lemma $\Rightarrow$ connecting homomorphism
and
nonabelian homological algebra
A basic lemma in homological algebra: it constructs connecting homomorphisms.
Let
be a commuting diagram in an abelian category $\mathcal{A}$ such that the two rows are exact sequences.
Then there is a long exact sequence of kernels and cokernels of the form
Moreover
if $A \to B$ is a monomorphism then so is $ker(f) \to ker(g)$
if $B \to C$ is an epimorphism, then so is $coker(g) \to coker(h)$.
If $\mathcal{A}$ is realized as a (full subcategory of) a category of $R$-modules, then the connecting homomorphism $\partial$ here can be defined on elements $c' \in ker(h) \subset C'$ by
where $i^{-1}(-)$ and $p^{-1}(-)$ denote any choice of pre-image (the total formula is independent of that choice).
The snake lemma derives its name from the fact that one may draw the connecting homomorphism $\partial$ that it constructs diagrammatically as follows:
An early occurence of the snake lemma is as lemma (5.8) of
In
it appears as lemma 1.3.2.
A purely category-theoretic proof is given in
and in
See also