Certainly Bourbaki never wrote an introduction to algebraic geometry: we would have heard about it, right?

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    $\begingroup$ By the way, I think this is an awesome example of answering your own question in one go. We don't see that much on MO, and I think that is fine. But in this case: well done! $\endgroup$ – jmc Mar 20 '15 at 10:00
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    $\begingroup$ Thank you for your encouragement, @jmc: I was a bit hesitant before posting both question and answer, which I had never done before. $\endgroup$ – Georges Elencwajg Mar 20 '15 at 10:11
  • $\begingroup$ I think it's the exceptional general interest of the question which is crucial. If the question and answer did not have that character -- as if the author were talking to him/herself -- the reception could be different. $\endgroup$ – Todd Trimble May 29 '16 at 23:25
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    $\begingroup$ I remember my advisor, John Tate, telling me back in the 1970s that Lie Algebras was the perfect topic of a series of Bourbaki books, unlike algebraic geometry which is far too vast. $\endgroup$ – Dan Flath May 30 '17 at 20:19
  • $\begingroup$ @DanFlath Welcome to MO, Professor Flath. I converted your answer to a comment; we are pretty strict around here about reserving answer boxes for responses that directly answer the question. One you gain 50 points of "reputation", you can leave comments on any post. $\endgroup$ – Todd Trimble May 30 '17 at 21:02

Here is Bourbaki document on algebraic geometry, taken from the now available Master's Archives: click on Autres rédactions, then on Chap.I Théorie globale élémentaire (91 p.)

This preliminary draft was apparently written (according to a penciled annotation on the first page) for Bourbaki in 1954 by Samuel, a distinguished algebraic geometer and number theorist.
Alas, it is hard to conceive a worse timing for a book on algebraic geometry: one year later Serre would publish his paradigm shifting FAC, shortly followed by Grothendieck's theory of schemes, a vast development of Serre's article (as acknowledged in the Preface to the EGA), which would forever change our vision of algebraic geometry.
Samuel's point of view is that of Weil: at the forefront is a "universal domain", a field extension $k\subset K$ with $K$ algebraically closed and of infinite transcendency degree over $k$.

Geometry would happen in $\mathbb A^n(K)$ or $\mathbb P^n(K)$, whereas algebra and number theory would take place inside $k[T_1,\dots,T_n]$ or $k[T_0,\dots,T_n]$.
A variety in Weil's vision could have a multitude of generic points, essentially points such that a polynomial vanishing on them must be zero.
It is quite moving to see the author struggling with, for example, the product of varieties: he notices the difficulty due to the tensor product $E\otimes_k F$ of two field extensions $k\subset E,F$ having non-zero nilpotents but doesn't envision incorporating these in his foundational text.
Grothendieck would soon show the world how considering nilpotents in the very foundations of scheme theory would enrich and beautify algebraic geometry.

I encourage every algebraic geometer to browse this nostalgic and unacknowledged witness of a bygone era of our beloved science.

Edit (May 27th, 2016)
Browsing the fascinating Grothendieck-Serre Correspondence ( a review of which is here) I found this excerpt from the very first letter of the Correspondence (page 3, dated January 28th, 1955), written by Grothendieck then in Lawrence, Kansas, USA :

"You said that Bourbaki wanted to send me a draft by Samuel on algebraic geometry (and commutative algebra?). I would be happy to get it..."

This confirms that the document mentioned above was indeed authored by Samuel.

  • $\begingroup$ Georges, thanks for an interesting post. Did Grothendieck try to publish his vision of algebraic geometry in the context of the Bourbaki oeuvre? Is there a discussion of this in the literature? $\endgroup$ – Mikhail Katz Jun 1 '17 at 8:18
  • $\begingroup$ Dear @Mikhail, Grothendieck did not try to publish his vision of algebraic geometry in the Bourbaki series "Éléments de Mathématique" because it would have taken several thousands of pages. However he published several surveys in the "Séminaire Bourbaki", most notably the one pertaining to the construction of the Hilbert schemes: see here. $\endgroup$ – Georges Elencwajg Jun 1 '17 at 17:25
  • $\begingroup$ Georges, thanks. With regard to this text by Samuel, did I understand you correctly that the text was proposed to Bourbaki but not published? Is this text by Samuel mentioned in any publication and/or its history discussed? $\endgroup$ – Mikhail Katz Jun 5 '17 at 16:23
  • $\begingroup$ @Mikhail: Bourbaki used to ask his members to write preliminary reports/drafts on many subjects and I guess Samuel's text was a very preliminary such draft. From what I understand these drafts went through many stages before being eventually incorporated in the official "Éléments de Mathématique". Many books however (like Godement's "Théorie des faisceaux" or Lang's "Rapport sur la cohomologie des groupes" ) originated from drafts that had not been so incorporated. The only place I know where Samuel's text is evoked is in Grothendieck's letter to Serre mentioned in the Edit to my answer. $\endgroup$ – Georges Elencwajg Jun 5 '17 at 18:24
  • $\begingroup$ Georges, I have a vague memory of seeing a reference to such a text in I think one of Adrian Mathias's articles on the Bourbaki but I can't quite place it. If you come across it please let me know. $\endgroup$ – Mikhail Katz Jun 6 '17 at 7:59

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