Short exact sequence

Split short exact sequence

A split short exact sequence¹ is a short exact sequence (depicted above) in an Abelian category that is equivalent to

which is always exact.

Equivalent characterizations

The following characterisations are equivalent:² homology

1. the sequence splits;
2. is a split epimorphism;
3. is a split monomorphism.

Proof

We prove for a sequence in and thus for any Abelian category via Freyd-Mitchell theorem.

Consider a split sequence, i.e. the following diagram commutes.

has a right-inverse, namely for . Then , so is a right-inverse of . Therefore 1. implies 2.

Now take a short sequence such that has a right-inverse with . Since is injective, there exists an inverse on its range . for all since

Thus we may define , which is a left-inverse of since

for all . Therefore 2. implies 3..

Finally take a short sequence such that has a left-inverse with . Let . Then is a morphism of short exact sequences, since

and

Hence by the Five lemma is an isomorphism, whence is an isomorphism of short sequences. Therefore 3. implies 1..

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Footnotes

1. German spaltete kurze exakte Sequenz ↩

2. 2010, Algebraische Topologie, ¶3.1.11, pp. 132ff ↩