Splitting lemma

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence

0\longrightarrow A\mathrel {\overset {q}{\longrightarrow }} B\mathrel {\overset {r}{\longrightarrow }} C\longrightarrow 0.

Left split
There exists a morphism $t : B \to A$ such that $tq$ is the identity $id A$ on $A$ ,
Right split
There exists a morphism $u : C \to B$ such that $ru$ is the identity $id C$ on $C$ ,
Direct sum
There is an isomorphism $h$ from $B$ to the direct sum of $A$ and $C$ , such that $hq$ is the natural injection of $A$ into the direct sum, and $rh^{-1}$ is the natural projection of the direct sum onto $C$ .

If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to split.

In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that:

C ≅ B /ker r ≅ B / q (A)

(i.e.,

C

isomorphic to the coimage of

r

q

)

to:

B = q (A) \oplus u (C) ≅ A \oplus C

where the first isomorphism theorem is then just the projection onto $C$ .

It is a categorical generalization of the rank–nullity theorem (in the form $V ≅ ker T \oplus im T)$ in linear algebra.