nLab split exact sequence



Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




In an abelian category

Let π’œ\mathcal{A} be an abelian category.


A short exact sequence 0β†’Aβ†’iBβ†’pCβ†’00\to A \stackrel{i}{\to} B \stackrel{p}{\to} C\to 0 in π’œ\mathcal{A} is called split if either of the following equivalent conditions hold

  1. There exists a section of pp, hence a morphism s:Cβ†’Bs \colon C\to B such that p∘s=id Cp \circ s = id_C.

  2. There exists a retract of ii, hence a morphism r:Bβ†’Ar \colon B\to A such that r∘i=id Ar \circ i = id_A.

  3. There exists an isomorphism of sequences with the sequence

    0β†’Aβ†’AβŠ•Cβ†’Cβ†’0 0\to A\to A\oplus C\to C\to 0

    given by the direct sum and its canonical injection/projection morphisms.


(splitting lemma)

The three conditions in def. are indeed equivalent.

(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 r:B→Ar \colon B \to A of i:A→Bi \colon A \to B. Write P:B→rA→iBP \colon B \stackrel{r}{\to} A \stackrel{i}{\to} B for the corresponding idempotent.

Then every element b∈Bb \in B can be decomposed as b=(bβˆ’P(b))+P(b)b = (b - P(b)) + P(b) hence with bβˆ’P(b)∈ker(r)b - P(b) \in ker(r) and P(b)∈im(i)P(b) \in im(i). Moreover this decomposition is unique since if b=i(a)b = i(a) while at the same time r(b)=0r(b) = 0 then 0=r(i(a))=a0 = r(i(a)) = a. This shows that B≃im(i)βŠ•ker(r)B \simeq im(i) \oplus ker(r) is a direct sum and that i:Aβ†’Bi \colon A \to B is the canonical inclusion of im(i)im(i). By exactness it then follows that ker(r)≃im(p)ker(r) \simeq im(p) and hence that B≃AβŠ•CB \simeq A \oplus C with the canonical inclusion and projection.

The implication that the second condition also implies the third is formally dual to this argument.

In a semi-abelian category

There is a nonabelian analog of split exact sequences in semiabelian categories. See there.


Relation to chain homotopy


A long exact sequence C β€’C_\bullet is split exact precisely if the weak homotopy equivalence from the 0-chain complex, namely the quasi-isomorphism 0β†’C β€’0 \to C_\bullet is actually a chain homotopy equivalence, in that the identity on C β€’C_\bullet has a null homotopy.

Of free modules and vector spaces

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 kk be a field and denote by π’œβ‰”k\mathcal{A} \coloneqq kVect the category of vector spaces over kk.


Every short exact sequence of vector spaces is split.

(Essentially by the basis theorem, for exposition see for instance here.)

Involving injective/projective objects


If in a short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 in an abelian category the first object AA 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 A→BA \to B here is a monomorphism. By definition of injective object, if AA is injective then it has the right lifting property against monomorphisms and so there is a morphism q:B→Aq : B \to A that makes the following diagram commute:

A β†’id A A ↓ β†— q B. \array{ A &\stackrel{id_A}{\to}& A \\ \downarrow & \nearrow_{q} \\ B } \,.

Hence qq is a retract as in def. .


For instance

Last revised on April 23, 2023 at 09:35:45. See the history of this page for a list of all contributions to it.