Definition 26.6.1. Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ S$-module1. The vector bundle associated to $\mathcal{E}$ is

$\mathbf{V}(\mathcal{E}) = \underline{\mathop{\mathrm{Spec}}}_ S(\text{Sym}(\mathcal{E})).$
[1] The reader may expect here the condition that $\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with [II, Definition 1.7.8, EGA].

Comment #4182 by Dmitrii Pedchenko on

In complex analytic geometry there is a tradition of calling the resulting geometric object for a non-locally free sheaf a "linear fiber space over $S$". (See, for instance, G. Fischer "Complex analytic geometry", p. 50, or Grauert, et. al "Several complex variables VII", p. 119). M. Atiyah called such objects "families of vector spaces" in this K-theory work.

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