# Picard group

The Picard group^{} of a variety, scheme, or more generally locally
ringed space $(X,{O}_{X})$ is the group of locally free ${O}_{X}$ modules of rank
$1$ with tensor product^{} over ${O}_{X}$ as the operation, usually denoted by $\mathrm{Pic}(X)$. Alternatively, the Picard group is the group of isomorphism classes of invertible sheaves on $X$, under tensor products.

It is not difficult to see that $\mathrm{Pic}(X)$ is isomorphic to ${\mathrm{H}}^{1}(X,{O}_{X}^{*})$, the first sheaf cohomology group of the multiplicative sheaf ${O}_{X}^{*}$ which consists of the units of ${O}_{X}$.

Finally, let $\mathrm{CaCl}(X)$ be the group of Cartier divisors on $X$ modulo linear equivalence. If $X$ is an integral scheme then the groups $\mathrm{CaCl}(X)$ and $\mathrm{Pic}(X)$ are isomorphic. Furthermote, if we let $\mathrm{Cl}(X)$ be the class group^{} of Weil divisors (divisors^{} modulo principal divisors) and $X$ is a noetherian^{}, integral and separated locally factorial scheme, then there is a natural isomorphism $\mathrm{Cl}(X)\cong \mathrm{Pic}(X)$. Thus, the Picard group is sometimes called the divisor class group of $X$.

Title | Picard group |
---|---|

Canonical name | PicardGroup |

Date of creation | 2013-03-22 12:52:30 |

Last modified on | 2013-03-22 12:52:30 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 6 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 14-00 |

Synonym | divisor class group |