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Nov 13, 2009 at 6:12 comment added Csar Lozano Huerta Some additional comment is the following. The type of vector bundles you are considering above are algebraic bundles. To give a sense that $\mathcal{C}^{\infty}$-bundles may be very distinct objects from those that you are considering above, here is a result due to Quillen (1978) Algebraic vector bundles over $\mathcal{C}^{\infty}$ are trivial. this result is trivial on the topological category.
Nov 6, 2009 at 17:19 history edited Ilya Nikokoshev CC BY-SA 2.5
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Oct 22, 2009 at 17:54 comment added Ilya Nikokoshev Here's a bonus: I was thinking about good books that would emphasize this connection between commutative algebra and schemes and the name 'Eisenbud' came to my mind.
Oct 22, 2009 at 13:59 vote accept Qiaochu Yuan
Oct 22, 2009 at 8:45 history edited Ilya Nikokoshev CC BY-SA 2.5
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Oct 22, 2009 at 8:37 history answered Ilya Nikokoshev CC BY-SA 2.5