The Salamander Lemma

For each piece of the double complex of the form

we obtain the following salamander-shaped diagram of mural maps:

If a diagram

is part of a double complex in an abelian category, then there is a 6-term exact sequence

A_{◻} \to B^{hor} \to B_{◻} \to^{◻} C \to C^{hor} \to^{◻} D,

where the first morphism is the composition $A_{◻} \to^{◻} B \to B^{hor}$ and the last morphism is the composition $C^{hor} \to C_{◻} \to^{◻} D$ .

Similarly, if a diagram

is part of a double complex in an abelian category, then there is a 6-term exact sequence

A_{◻} \to B^{vert} \to B_{◻} \to^{◻} C \to C^{vert} \to^{◻} D,

where the first morphism is the composition $A_{◻} \to^{◻} B \to B^{vert}$ and the last morphism is the composition $C^{vert} \to C_{◻} \to^{◻} D$ .

I'll include the proof of this at some point, but for now see here and notice how it's not very long.

Intramural and extramural isomorphisms

Extramural isomorphisms

If for some horizontal arrow $\partial^{hor} : A \to B$ in the double complex one has $A^{hor} = 0 = B^{hor}$ , then the extramural map $A_{◻} ↗^{◻} B$ is an isomorphism.

Similarly, if for some vertical arrow $\partial^{vert} : A \to B$ in the double complex one has $A^{vert} = 0 = B^{vert}$ , then the extramural map

\begin{aligned} A & _{◻} \\ ↙ \\ ^{◻} B \end{aligned}

is an isomorphism.

(A proof will be added here eventually.)

Intramural isomorphisms

If $\partial : A \to B$ is a morphism in a double complex and at one end of that morphism the complex is exact in the perpendicular direction, then two of the intramural maps are isomorphisms (as shown below; I'll TeX up this massive diagram eventually):

(A proof will be added here eventually.)