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 $X$ .