Schema/Lokal freie Garben und Vektorbündel/Äquivalenz/en/Fakt

Let $X$ denote a scheme.

Then the category of locally free sheaves on $X$ and the category of geometric vector bundles on $X$ are equivalent.

A geometric vector bundle $V \to X$ corresponds to the sheaf of its sections, and a locally free sheaf $ℱ$ corresponds to the (relative) spectrum of the symmetric algebra of the dual module $ℱ^{*}$ .