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 characterizationsEdit
A short exact sequence of abelian groups or of modules over a fixed 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 <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
The splitting lemma provides further equivalent characterizations of split exact sequences.
ExamplesEdit
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 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 <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 notionsEdit
Pure exact sequences can be characterized as the filtered colimits of split exact sequences.<ref>Template:Harvtxt</ref>