A sheaf $\L$ of $\O_X$ modules on a ringed space $\O_X$ is called invertible if there is another sheaf of $\O_X$ -modules $\L'$ such that $\L\otimes\L'\cong\O_X$ . A sheaf is invertible if and only if it is locally free of rank 1, and its inverse is the sheaf $\L^{\vee}\cong\mathcal{H}om(\L,\O_X)$ , by the obvious map.

The set of invertible sheaves obviously form an abelian group under tensor multiplication, called the Picard group of $X$ .