Editing Split exact sequence

{{Short description|Type of short exact sequence in mathematics}}

The term '''split exact sequence''' is used in two different ways by different people.  Some people mean a [[short exact sequence]] that right-splits (thus corresponding to a semidirect product) and some people mean a [[short exact sequence]] 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]], 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 group]]s or of [[module (mathematics)|modules]] over a fixed [[Ring (mathematics)|ring]], or more generally of objects in an [[abelian category]]

:<math>0 \to A \mathrel{\stackrel{a}{\to}} B \mathrel{\stackrel{b}{\to}} C \to 0</math>

is called split exact if it is isomorphic to the exact sequence where the middle term is the [[direct sum]] of the outer ones:

:<math>0 \to A \mathrel{\stackrel{i}{\to}} A \oplus C \mathrel{\stackrel{p}{\to}} C \to 0</math>

The requirement that the sequence is isomorphic means that there is an [[isomorphism]] <math>f : B \to A \oplus C</math> such that the composite <math>f \circ a</math> is the natural [[Inclusion map|inclusion]] <math>i: A \to A \oplus C</math> and such that the composite <math>p \circ f</math> equals ''b''. This can be summarized by a [[commutative diagram]] as:

[[File:Commutative diagram for split exact sequence - fixed.svg|frameless|311x311px]]

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

==Examples==
A trivial example of a split short exact sequence is 
:<math>0 \to M_1 \mathrel{\stackrel{q}{\to}} M_1\oplus M_2 \mathrel{\stackrel{p}{\to}} M_2 \to 0</math>
where <math>M_1, M_2</math> are ''R''-modules, <math>q</math> is the canonical injection and <math>p</math> is the canonical projection.
 
Any short exact sequence of [[vector space]]s is split exact. This is a rephrasing of the fact that any [[Set (mathematics)|set]] of [[linearly independent]] vectors in a vector space can be extended to a [[Basis (linear algebra)|basis]].

The exact sequence <math>0 \to \mathbf{Z}\mathrel{\stackrel{2}{\to}} \mathbf{Z}\to \mathbf{Z}/ 2\mathbf{Z} \to 0</math> (where the first map is multiplication by 2) is not split exact.

==Related notions==

[[Pure exact sequence]]s can be characterized as the [[filtered colimit]]s of split exact sequences.<ref>{{harvtxt|Fuchs|2015|loc=Ch. 5, Thm. 3.4}}</ref>

==References==
{{Reflist}}

==Sources==
*{{Citation|title=Abelian Groups|author=Fuchs|first=László|authorlink = László Fuchs|isbn=9783319194226|series=Springer Monographs in Mathematics|year=2015|publisher=Springer}}
*{{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}}
[[Category:Abstract algebra]]