invertible sheaf

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

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