# Cohen-Macaulay module

A module $M$ over a ring $R$ is a Cohen-Macaulay module if its depth is defined and equals its Krull dimension. A ring is said to be Cohen-Macaulay (or just C-M) if it is a Cohen-Macaulay module viewed as a module over itself.

Cohen-Macaulay rings are used extensively in combinatorial geometry and commutative ring theory, and has applications to algebraic geometry^{} as well. For instance, a variety all of whose local rings^{} are Cohen-Macaulay has, in a sense, nicer behaviour than an arbitrary singular variety.

Title | Cohen-Macaulay module |
---|---|

Canonical name | CohenMacaulayModule |

Date of creation | 2013-03-22 14:14:58 |

Last modified on | 2013-03-22 14:14:58 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 13C14 |

Classification | msc 16E65 |

Defines | Cohen-Macaulay ring |

Defines | C-M module |

Defines | C-M ring |