Stalk inclusion into rational sections of a locally free sheaf (${\mathcal{F}}_P\hookrightarrow \mathrm{Rat}(\mathcal{F})$)

On an irreducible variety, every nonempty open subset is dense, so a germ determines a rational section. For a locally free sheaf, this map is injective: a local section which vanishes on a dense open has all its regular coefficient functions zero, hence is zero. Local freeness is important; a section of a torsion sheaf can vanish generically without having zero germ.