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.
Martin Brandenburg
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4Dear 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. – Georges Elencwajg Jan 21 '14 at 09:58
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Oh, this wasn't intended :). G is next to H. – Martin Brandenburg Jan 21 '14 at 10:03
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Perfect! Thank you very much guys :) – AB_IM 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".
Georges Elencwajg
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1Thanks I almost forget how much of a pleasure it is to read from Serre :) – AB_IM Jan 21 '14 at 17:29
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2Dear CSA: ...and how much of a pleasure it is to read your so true comment ! – Georges Elencwajg Jan 21 '14 at 17:57
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1@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". – aytio Apr 10 '17 at 08:57
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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. – Georges Elencwajg Apr 10 '17 at 11:11