# Picard group

The 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 PicardGroup 2013-03-22 12:52:30 2013-03-22 12:52:30 alozano (2414) alozano (2414) 6 alozano (2414) Definition msc 14-00 divisor class group