# coherent sheaf

Let $R$ be a ring with unity, and $X=\mathrm{Spec}R$ be its prime spectrum. Given an $R$-module $M$, one can define a presheaf^{} on $X$ by defining its sections^{} on an open set $U$ to be ${\mathcal{O}}_{X}(U){\otimes}_{R}M$. We call the sheafification^{} of this $\stackrel{~}{M}$, and a sheaf of this form on $X$ is called quasi-coherent. If $M$ is a finitely generated module, then $\stackrel{~}{M}$ is called coherent. A sheaf on an arbitrary scheme $X$ is called (quasi-)coherent if it is (quasi-)coherent on each open affine subset of $X$.

Title | coherent sheaf |
---|---|

Canonical name | CoherentSheaf |

Date of creation | 2013-03-22 13:51:27 |

Last modified on | 2013-03-22 13:51:27 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 11 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 14A15 |

Synonym | quasi-coherent sheaf |