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 $\operatorname{Pic}(X)$ . 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 $\operatorname{Pic}(X)$ is isomorphic to ${\rm H}^{1}(X,O_{X}^{*})$ , the first sheaf cohomology group of the multiplicative sheaf $O_{X}^{*}$ which consists of the units of $O_{X}$ .

Finally, let $\operatorname{CaCl}(X)$ be the group of Cartier divisors on $X$ modulo linear equivalence. If $X$ is an integral scheme then the groups $\operatorname{CaCl}(X)$ and $\operatorname{Pic}(X)$ are isomorphic. Furthermote, if we let $\operatorname{Cl}(X)$ 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 $\operatorname{Cl}(X)\cong\operatorname{Pic}(X)$ . 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