Dual of a sheaf (${\mathcal{F}}^{\vee }=\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},{\mathcal{O}}_X)$)

The dual is the sheaf of modules whose sections on $U$ are module-sheaf morphisms $\mathcal{F}{|}_U\to {\mathcal{O}}_U$. For a locally free sheaf of finite rank, the dual is locally free of the same rank and the natural double-dual map is an isomorphism. General coherent sheaves need not have this property: a skyscraper at a point of positive-dimensional projective space has zero ordinary sheaf dual.