# exact sequence

If we have two homomorphisms^{} $f:A\to B$ and $g:B\to C$
in some category^{} of modules,
then we say that $f$ and $g$ are exact at $B$
if the image of $f$ is equal to the kernel of $g$.

A sequence^{} of homomorphisms

$$\mathrm{\cdots}\to {A}_{n+1}\stackrel{{f}_{n+1}}{\u27f6}{A}_{n}\stackrel{{f}_{n}}{\u27f6}{A}_{n-1}\to \mathrm{\cdots}$$ |

is said to be exact if each pair of adjacent homomorphisms $({f}_{n+1},{f}_{n})$ is exact – in other words if $\mathrm{im}{f}_{n+1}=\mathrm{ker}{f}_{n}$ for all $n$.

Compare this to the notion of a chain complex^{}.

Remark. The notion of exact sequences^{} can be generalized to any abelian category^{} $\mathcal{A}$, where ${A}_{i}$ and ${f}_{i}$ above are objects and morphisms in $\mathcal{A}$.

Title | exact sequence |

Canonical name | ExactSequence |

Date of creation | 2013-03-22 12:09:27 |

Last modified on | 2013-03-22 12:09:27 |

Owner | antizeus (11) |

Last modified by | antizeus (11) |

Numerical id | 7 |

Author | antizeus (11) |

Entry type | Definition |

Classification | msc 16-00 |

Related topic | ExactSequence2 |

Related topic | CategoricalSequence |

Related topic | HomologicalComplexOfTopologicalVectorSpaces |

Related topic | CategoricalDiagramsAsFunctors |

Related topic | SpinGroup |

Related topic | AlternativeDefinitionOfAnAbelianCategory |