# construction of an injective resolution

The category^{} of modules has enough injectives^{}.
Let $M$ be a module, and let ${I}^{0}$ be an injective module^{} such that

$$0\u27f6M\u27f6{I}^{0}$$ |

is exact. Then, let ${M}_{0}$ be the image of $M$ in ${I}^{0}$, and construct the factor module ${I}^{0}/{M}^{0}$. Then, since the category of modules has enough injectives, we can find a module ${I}^{1}$ such that

$$0\u27f6{I}^{0}/{M}^{0}\stackrel{{\varphi}_{0}}{\u27f6}{I}^{1}$$ |

is exact. ${\varphi}_{0}$ induces a homomorphism^{} $\varphi :{I}^{0}\u27f6{I}^{1}$, whose kernel is ${M}^{0}$. We thus have an exact sequence^{}

$$0\u27f6M\u27f6{I}^{0}\u27f6{I}^{1}.$$ |

One can continue this process to construct injective modules ${I}^{n}$ for any $n\in \mathbb{Z}$ (the resolution may terminate: ${I}^{m}=0$ for some $N\in \mathbb{Z}$ with all $m>N$).

Title | construction of an injective resolution |
---|---|

Canonical name | ConstructionOfAnInjectiveResolution |

Date of creation | 2013-03-22 17:11:02 |

Last modified on | 2013-03-22 17:11:02 |

Owner | guffin (12505) |

Last modified by | guffin (12505) |

Numerical id | 5 |

Author | guffin (12505) |

Entry type | Derivation^{} |

Classification | msc 16E05 |