# 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 |