The Stacks project

Definition 27.6.1. 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].

Comments (2)

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 ". (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 on

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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  • 6 comment(s) on Section 27.6: Vector bundles

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View Definition 27.6.1 as pdf