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What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?

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Chapter II, §5.2 in:

N. Bourbaki, Elements of mathematics. Commutative algebra. Hermann, Paris, 1972. Translated from the French.

Alternatively, you can always consult the stacks project for these basics. In this case, it's Tag 00NV.

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    $\begingroup$ Dear Martin: without any disrespect to your country, the publisher of the Bourbaki volume you mention is Hermann, not Germann. Apart from that I can only second you on this excellent reference. $\endgroup$ – Georges Elencwajg Jan 21 '14 at 9:58
  • $\begingroup$ Oh, this wasn't intended :). G is next to H. $\endgroup$ – Martin Brandenburg Jan 21 '14 at 10:03
  • $\begingroup$ Perfect! Thank you very much guys :) $\endgroup$ – AIM_BLB Jan 21 '14 at 17:19
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Why not read Serre's crystal-clear article which introduced this equivalence?
As Gauss said: "Read the masters".

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    $\begingroup$ Thanks I almost forget how much of a pleasure it is to read from Serre :) $\endgroup$ – AIM_BLB Jan 21 '14 at 17:29
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    $\begingroup$ Dear CSA: ...and how much of a pleasure it is to read your so true comment ! $\endgroup$ – Georges Elencwajg Jan 21 '14 at 17:57
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    $\begingroup$ @GeorgesElencwajg, sorry for commenting on an old thread, but the link to the paper is broken. I believe this is the paper Georges had in mind. In case the link breaks again, the name of the article is "Modules projectifs et espaces fibres a fibre vectorielle". $\endgroup$ – aytio Apr 10 '17 at 8:57
  • $\begingroup$ Thanks @aytio: that's indeed the article, from Séminaire Dubreil 1957-1958. I have modified my link. Just to be on the safe side, here is a different link to the same article. And yet another one. $\endgroup$ – Georges Elencwajg Apr 10 '17 at 11:11

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