# locally free

A sheaf of ${\mathcal{O}}_{X}$-modules $\mathcal{F}$ on a ringed space $X$ is called locally free if for each point $p\in X$, there is an open neighborhood (http://planetmath.org/Neighborhood^{})
$U$ of $x$ such that ${\mathcal{F}|}_{U}$ is free (http://planetmath.org/FreeModule) as an ${{\mathcal{O}}_{X}|}_{U}$-module, or equivalently, ${\mathcal{F}}_{p}$, the stalk of $\mathcal{F}$ at $p$, is free as a ${({\mathcal{O}}_{X})}_{p}$-module. If ${\mathcal{F}}_{p}$ is of finite rank (http://planetmath.org/ModuleOfFiniteRank) $n$, then $\mathcal{F}$ is said to be of rank $n$.

Title | locally free |
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Canonical name | LocallyFree |

Date of creation | 2013-03-22 13:52:31 |

Last modified on | 2013-03-22 13:52:31 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 13 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 14A99 |