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Concept# Splitting lemma

Summary

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any , 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.
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 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 generalization of the rank–nullity theorem (in the form V ≅ ker T ⊕ im T) in linear algebra.
Proof for the category of abelian groups

- ⇒ 1. and 3. ⇒ 2. First, to show that 3. implies both 1. and 2., we assume 3. a

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