(also nonabelian homological algebra)
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Let be an abelian category.
A short exact sequence in is called split if either of the following equivalent conditions hold
There exists a section of , hence a morphism such that .
There exists a retract of , hence a morphism such that .
There exists an isomorphism of sequences with the sequence
given by the direct sum and its canonical injection/projection morphisms.
(e.g. Hatcher (2002), p. 147)
It is clear that the third condition implies the first two: take the section/retract to be given by the canonical injection/projection maps that come with a direct sum.
Conversely, suppose we have a retract of . Write for the corresponding idempotent.
Then every element can be decomposed as hence with and . Moreover this decomposition is unique since if while at the same time then . This shows that is a direct sum and that is the canonical inclusion of . By exactness it then follows that and hence that with the canonical inclusion and projection.
The implication that the second condition also implies the third is formally dual to this argument.
There is a nonabelian analog of split exact sequences in semiabelian categories. See there.
A long exact sequence is split exact precisely if the weak homotopy equivalence from the 0-chain complex, namely the quasi-isomorphism is actually a chain homotopy equivalence, in that the identity on has a null homotopy.
Assuming the axiom of choice:
Every exact sequence of free abelian groups is split.
Every exact sequence of free modules which is bounded below is split.
Let be a field and denote by Vect the category of vector spaces over .
Every short exact sequence of vector spaces is split.
(Essentially by the basis theorem, for exposition see for instance here.)
If in a short exact sequence in an abelian category the first object is an injective object or the last object is a projective object then the sequence is split exact.
Consider the first case. The other is formally dual.
By the properties of a short exact sequence the morphism here is a monomorphism. By definition of injective object, if is injective then it has the right lifting property against monomorphisms and so there is a morphism that makes the following diagram commute:
Hence is a retract as in def. .
For instance
Charles Weibel, Section 1.4 of: An Introduction to Homological Algebra (1994)
Allen Hatcher, pp. 147 of: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]
Last revised on April 23, 2023 at 09:35:45. See the history of this page for a list of all contributions to it.