mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a split exact sequence is a

short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...

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 group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...

s or of

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...

over a fixed

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

, 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 of ab ...

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. To see how the direct sum is used in abstract algebra, consider a more ...

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 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 them. The word is ...

f : B \to A \oplus C

such that the composite

f \circ a

is the natural

inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...

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 In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence : 0 \longrightarrow A \mathrel B \mathrel C \longrightarrow ...

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 whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

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

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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

vectors in a vector space can be extended to a

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

. The exact sequence

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

(where the first map is multiplication by 2) is not split exact.

Related notions

Pure exact sequence In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generali ...

s can be characterized as the

filtered colimit In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered c ...

s 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