Five lemma

Remarks

The lemma is the conjunction of the two halves of the four lemma. The left four columns, with $m, p$ monomorphisms and $l$ an epimorphism, show that $n$ is a monomorphism. The right four columns, with $m, p$ epimorphisms and $q$ a monomorphism, show that $n$ is an epimorphism. Each half is a diagram chase. A typical use: a map between two short exact sequences that is an isomorphism on the outer terms is an isomorphism on the middle term, by taking $A = A' = E = E' = 0$. The existence of the map $n$ making the diagram commute is essential: $0 \to \Z/2\Z \to \Z/4\Z \to \Z/2\Z \to 0$ and $0 \to \Z/2\Z \to \Z/2\Z \oplus \Z/2\Z \to \Z/2\Z \to 0$ have isomorphic outer terms, but $\Z/4\Z \not\cong \Z/2\Z \oplus \Z/2\Z$.

Going deeper

For a student

The five lemma is used constantly to show that a natural map is an isomorphism. One fits the map into a ladder of long exact sequences, for example in homology, where the other maps are already known to be isomorphisms. This is how one proves that a map of spaces that induces isomorphisms on the homology of pieces induces an isomorphism on the whole, as in Mayer–Vietoris arguments. It also proves the splitting lemma and the invariance of derived functors under the choice of resolution.