split short exact sequence
SplitShortExactSequence
2002-01-05 15:10:59
2003-09-20 21:44:37
Definition
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In an abelian category, a short exact sequence
$0 \to A \buildrel f \over \to B \buildrel g \over \to C \to 0$
is {\it 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 {\it backmaps} or {\it splitting backmaps}.