Picard group

The Picard group of a variety, scheme, or more generally locallyringed 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 ${\mathrm{H}}^1(X,O_X^{*})$, thefirst sheaf cohomology group of the multiplicative sheaf $O_X^{*}$ which consists of theunits 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 $X$.