Definition 27.6.1 (01M2)—The Stacks project

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Definition 27.6.1. Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ S$ -module ¹. 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].

Comments (2)

Comment #4182 by Dmitrii Pedchenko on April 21, 2019 at 16:00

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.

Comment #4378 by Johan on August 30, 2019 at 08:10

Interesting, I didn't know that. For now let's keep the same language as in EGA even though it is occasionally perhaps a bit misleading.

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