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

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

1. Left split

   There exists a morphism t: B → A such that tq is the identity on A, id_A,

2. Right split

   There exists a morphism u: C → B such that ru is the identity on C, id_C,

3. 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 ${\displaystyle rh^{-1}}$ 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 or cokernel of q)

to:

B = q(A) ⊕ u(C) ≅ A ⊕ 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 ⊕ im T) in linear algebra.