split short exact sequence

In an abelian category, a short exact sequence $0\to A\buildrel f\over{\to}B\buildrel g\over{\to}C\to 0$ 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 SplitShortExactSequence 2013-03-22 12:09:32 2013-03-22 12:09:32 antizeus (11) antizeus (11) 8 antizeus (11) Definition msc 16E05 backmap splitting backmap