split short exact sequence

In an abelian category, a short exact sequence $0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0$ is split if it satisfies the following equivalent conditions:

(a) there exists a homomorphism $h:C\to B$ such that $gh=1_C$;

(b) there exists a homomorphism $j:B\to A$ such that $jf=1_A$;

(c) $B$ is isomorphic to the direct sum $A\oplus C$.

In this case, we say that $h$ and $j$ are backmaps or splitting backmaps.

Title	split short exact sequence
Canonical name	SplitShortExactSequence
Date of creation	2013-03-22 12:09:32
Last modified on	2013-03-22 12:09:32
Owner	antizeus (11)
Last modified by	antizeus (11)
Numerical id	8
Author	antizeus (11)
Entry type	Definition
Classification	msc 16E05
Synonym	backmap
Synonym	splitting backmap