The term split exact sequence is used in two different ways by different people. Some people mean a

short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...

that right-splits (thus corresponding to a semidirect product) and some people mean a

that left-splits (which implies it right-splits, and corresponds to a direct product). This article takes the latter approach, but both are in common use. When reading a book or paper, it is important to note precisely which of the two meanings is in use. In

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a split exact sequence is a

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 In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category o ...

0 \to A \mathrel B \mathrel C \to 0

is called split exact if it is isomorphic to the exact sequence where the middle term is the

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...

of the outer ones: :

0 \to A \mathrel A \oplus C \mathrel C \to 0

The requirement that the sequence is isomorphic means that there is an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

f : B \to A \oplus C

such that the composite

f \circ a

is the natural inclusion

i: A \to A \oplus C

and such that the composite

p \circ f

equals ''b''. This can be summarized by a

commutative diagram 350px, The commutative diagram used in the proof of the five lemma In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...

as:

Commutative diagram for split exact sequence - fixed

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

Examples

A trivial example of a split short exact sequence is :

0 \to M_1 \mathrel M_1\oplus M_2 \mathrel M_2 \to 0

where

M_1, M_2

are ''R''-modules,

q

is the canonical injection and

p

is the canonical projection. Any short exact sequence of

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

s is split exact. This is a rephrasing of the fact that any set of

linearly independent In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...

vectors in a vector space can be extended to a basis. The exact sequence

0 \to \mathbf\mathrel \mathbf\to \mathbf/ 2\mathbf \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.

References

Sources

* *{{Citation, title= Steps in Commutative Algebra, 2nd ed., author=Sharp, R. Y., first=Rodney, isbn=0521646235, series=London Mathematical Society Student Texts, year=2001, publisher=Cambridge University Press Abstract algebra