Horseshoe lemma

Remarks

The proof builds $P_\bullet$ one degree at a time. Since $P''_0$ is projective, the map $P''_0 \to A''$ lifts through $\pi$ to a map $P''_0 \to A$. Together with $P'_0 \to A' \to A$ this gives $P'_0 \oplus P''_0 \to A$, which is an epimorphism by the snake lemma. The kernels again form a short exact sequence, and the same step is repeated with them. The differential of $P_\bullet$ is in general not the direct sum of the differentials: it has an extra component $P''_n \to P'_{n-1}$. For example, over $\Z$ take $0 \to \Z \xrightarrow{\cdot 2} \Z \to \Z/2\Z \to 0$ with $P'_\bullet = \Z$ and $P''_\bullet = (\Z \xrightarrow{\cdot 2} \Z)$; the lemma gives a resolution of $\Z$ by $\Z^2$ in degree $0$ and $\Z$ in degree $1$. The dual statement holds for injective resolutions.

Going deeper

For a student

The lemma is the key step to show that a short exact sequence gives a long exact sequence of left derived functors. Applying an additive functor to the split sequences $0 \to P'_n \to P_n \to P''_n \to 0$ keeps them exact, and the long exact sequence in homology does the rest. The same construction, applied degree by degree, is used to build Cartan–Eilenberg resolutions.