The Stacks project

Definition 111.40.7. Let $X$ be a locally ringed space. The Picard group of $X$ is the set $\mathop{\mathrm{Pic}}\nolimits (X)$ of isomorphism classes of invertible $\mathcal{O}_ X$-modules with addition given by tensor product. See Modules, Definition 17.25.9. For a ring $R$ we set $\mathop{\mathrm{Pic}}\nolimits (R) = \mathop{\mathrm{Pic}}\nolimits (\mathop{\mathrm{Spec}}(R))$.