split short exact sequence

In an abelian categoryMathworldPlanetmathPlanetmathPlanetmath, a short exact sequenceMathworldPlanetmathPlanetmath 0AfBgC0 is split if it satisfies the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath conditions:

(a) there exists a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath h:CB such that gh=1C;

(b) there exists a homomorphism j:BA such that jf=1A;

(c) B is isomorphic to the direct sumMathworldPlanetmathPlanetmathPlanetmath AC.

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