The Stacks project

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].

Comments (1)

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.

There are also:

  • 2 comment(s) on Section 26.6: Vector bundles

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01M2. Beware of the difference between the letter 'O' and the digit '0'.