# invertible sheaf

A sheaf $\U0001d50f$ of ${\mathcal{O}}_{X}$ modules on a ringed space ${\mathcal{O}}_{X}$ is called if there is another sheaf of ${\mathcal{O}}_{X}$-modules ${\U0001d50f}^{\prime}$ such that $\U0001d50f\otimes {\U0001d50f}^{\prime}\cong {\mathcal{O}}_{X}$. A sheaf is invertible^{} if and only if it is locally free of rank 1, and its inverse^{} is the sheaf ${\U0001d50f}^{\vee}\cong \mathscr{H}om(\U0001d50f,{\mathcal{O}}_{X})$, by the map.

The set of invertible sheaves form an abelian group^{} under tensor multiplication, called the Picard group^{} of $X$.

Title | invertible sheaf |
---|---|

Canonical name | InvertibleSheaf |

Date of creation | 2013-03-22 13:52:34 |

Last modified on | 2013-03-22 13:52:34 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 8 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 14A99 |