In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.
Equivalent characterizations A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category
0 → A → a B → b C → 0 {\displaystyle 0\to A\mathrel {\stackrel {a}{\to }} B\mathrel {\stackrel {b}{\to }} C\to 0}
is called split exact if it is isomorphic to the exact sequence where the middle term is the direct sum of the outer ones:
0 → A → i A ⊕ C → p C → 0 {\displaystyle 0\to A\mathrel {\stackrel {i}{\to }} A\oplus C\mathrel {\stackrel {p}{\to }} C\to 0}
with i : A → A ⊕ C {\displaystyle i:A\to A\oplus C} being the natural inclusion of A into the direct sum, and p : A ⊕ C → C {\displaystyle p:A\oplus C\to C} denoting the natural projection of the direct sum onto the second summand. The requirement that the sequence is isomorphic means that there is an isomorphism f : B → A ⊕ C {\displaystyle f:B\to A\oplus C} such that the composite f ∘ a {\displaystyle f\circ a} is the natural inclusion i : A → A ⊕ C {\displaystyle i:A\to A\oplus C} and such that the composite p ∘ f {\displaystyle p\circ f} equals b. This can be summarized by a commutative diagram as:
The splitting lemma provides further equivalent characterizations of split exact sequences. The sequence
0 → A → a B → b C → 0 {\displaystyle 0\to A\mathrel {\stackrel {a}{\to }} B\mathrel {\stackrel {b}{\to }} C\to 0}
is split exact if and only if there exists r : C → B {\displaystyle r:C\to B} such that b ∘ r = 1 C {\displaystyle b\circ r=1_{C}} , which is the case if and only if there exists s : B → A {\displaystyle s:B\to A} such that s ∘ a = 1 A {\displaystyle s\circ a=1_{A}} .
Examples A trivial example of a split short exact sequence is
0 → M 1 → q M 1 ⊕ M 2 → p M 2 → 0 {\displaystyle 0\to M_{1}\mathrel {\stackrel {q}{\to }} M_{1}\oplus M_{2}\mathrel {\stackrel {p}{\to }} M_{2}\to 0}
where M 1 , M 2 {\displaystyle M_{1},M_{2}} are R-modules, q {\displaystyle q} is the canonical injection and p {\displaystyle p} is the canonical projection. Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis. The exact sequence 0 → Z → 2 Z → Z / 2 Z → 0 {\displaystyle 0\to \mathbf {Z} \mathrel {\stackrel {2}{\to }} \mathbf {Z} \to \mathbf {Z} /2\mathbf {Z} \to 0} (where the first map is multiplication by 2) is not split exact.
Related notions Pure exact sequences can be characterized as the filtered colimits of split exact sequences.
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