Splitting Lemma

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

Given a short exact sequence with maps q and r:

one writes the additional arrows t and u for maps that may not exist:

Then the following are equivalent (see Proposition 4.3, p.16, of the reference given below):

1. left split

there exists a map t: B → A such that tq is the identity on A,

2. right split

there exists a map u: C → B such that ru is the identity on C,

3. direct sum

B is isomorphic to the direct sum of A and C, with q corresponding to the natural injection of A and r corresponding to the natural projection onto C.

The short exact sequence is called split if any of the above statements hold.

(The word "map" refers to morphisms in the abelian category we are working in, not mappings between sets.)

It allows one to refine the first isomorphism theorem:

• the first isomorphism theorem states that in the above short exact sequence,
• if the sequence splits, then, and the first isomorphism theorem is just the projection onto C.

It is a categorical generalization of the rank–nullity theorem (in the form ) in linear algebra.