Picard group

The Picard groupMathworldPlanetmath of a variety, scheme, or more generally locally ringed space (X,OX) is the group of locally free OX modules of rank 1 with tensor productPlanetmathPlanetmath over OX as the operation, usually denoted by 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 Pic(X) is isomorphic to H1(X,OX*), the first sheaf cohomology group of the multiplicative sheaf OX* which consists of the units of OX.

Finally, let CaCl(X) be the group of Cartier divisors on X modulo linear equivalence. If X is an integral scheme then the groups CaCl(X) and Pic(X) are isomorphic. Furthermote, if we let Cl(X) be the class groupMathworldPlanetmath of Weil divisors (divisorsMathworldPlanetmathPlanetmath modulo principal divisors) and X is a noetherianPlanetmathPlanetmath, integral and separated locally factorial scheme, then there is a natural isomorphism Cl(X)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