Split exact sequence explained

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\toAl{\stackrel{a}{\to}}Bl{\stackrel{b}{\to}}C\to0

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\toAl{\stackrel{i}{\to}}A ⊕ Cl{\stackrel{p}{\to}}C\to0

f:B\toA ⊕ C

such that the composite

f\circa

is the natural inclusion

i:A\toA ⊕ C

and such that the composite

p\circf

equals b. This can be summarized by a commutative diagram as:

The splitting lemma provides further equivalent characterizations of split exact sequences.

Examples

A trivial example of a split short exact sequence is

0\toM₁l{\stackrel{q}{\to}}M_{1 ⊕}M₂l{\stackrel{p}{\to}}M₂\to0

where

M_1,M₂

are R-modules,

is the canonical injection and

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\toZl{\stackrel{2}{\to}}Z\toZ/2Z\to0

(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.