Invertible sheaves

Let $X$ be a scheme. An ${\mathcal{O}}_X$-module $\mathcal{L}$ is invertible if it is locally free of rank $1$ (Locally free sheaves of finite rank): every point $x\in X$ has an open neighbourhood $U$ with $\mathcal{L}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U\mathrm{.}$ Equivalently, $X$ is covered by open sets $U$ on which $\mathcal{L}{\mathrm{\mid }}_U$ admits a generator, that is, a section $s\in \mathcal{L}(U)$ such that the morphism ${\mathcal{O}}_U\to \mathcal{L}{\mathrm{\mid }}_U$, $a\mapsto a\cdot s{\mathrm{\mid }}_U$, is an isomorphism. The structure sheaf ${\mathcal{O}}_X$ is invertible.

Since local freeness of rank $1$ is a special case of local freeness of finite rank, an invertible sheaf is quasi-coherent, its rank function is the constant function $1$, the zero sheaf is not invertible on a nonempty scheme, and restriction to an open subscheme preserves invertibility. The tensor product, inverse and dual of invertible sheaves are treated on this page; in particular ${\mathcal{L}}^{\vee }=\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{L},{\mathcal{O}}_X)$ is again invertible of rank one, and ${\mathcal{L}}^{\vee }{\otimes }_{{\mathcal{O}}_X}\mathcal{L}$ is canonically ${\mathcal{O}}_X$.