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I would like this thread to contain possibly useful information about books and approaches on studying scheme theory for the first time. I'm truly sorry if one finds it inapproprite for this site, but from the experience I had with math.stackexchange I think that I've seen more "vague" threads.

Nowadays we have a massive number of introductions to modern algebraic geometry (scheme-theoretic, not complex, though in my humble opinion "algebraic geometry" is a bad name for the study of complex manifolds, "complex analytic geometry" describes the subject better) developed by Grothendieck and his school. I'll list the ones I know:

  • Ravi Vakil - "Foundations of Algebraic Geometry" (free on his website, very conversational and leaves a lot of stuff to exercises)

  • Robin Hartshorne - "Algebraic Geometry" (the classic, but studies only Noetherian schemes and possibly too formal for first times)

  • Ulrich Goertz and Torsten Wedhorn - "Algebraic Geometry: Part I: Schemes. With examples and exercises" (600 pages of scheme theory, but no cohomology which is reserved to yet unpublished second volume)

  • Joe Harris and David Eisenbud - "Geometry of Schemes" (introductory text on schemes, not a complete course on algebraic geometry, rather a text which tries to develop reader's intuition for studying schemes)

  • Kenji Ueno - "Algebraic Geometry 1/2/3" (was published in 1999 by AMS, but apparently not well known by western community as well as by me)

  • David Mumford and Tadao Oda - "Algebraic Geometry II" (expanded and updated version of Mumford's famous "Red book", seems neat and friendly)

  • Liu Qing - "Algebraic Geometry and Arithmetic Curves" (arithmetically flavoured text)

  • Alexander Grothendieck and Jean Dieudonne - "Elements of Algebraic Geometry" (the first "book" on algebraic geometry, very abstract and complete, 1800 pages-long, but exists only in French and possibly contains more than an beginner needs to know. But, maybe, even if all 1800 pages are not needed to learn scheme theory, they can be helpful to master it)

The (not algebraic, mind you) variety of books is definitely a good thing. But it can be daunting for a novice to choose which one to use? For example,

  • If one's inclinations lies in category theory and homological algebra/algebraic topology, which books emphasizes the "categorical" and "homological" modern approach? Introducing as much homological algebra as can be useful for the basics of scheme theory? (EGA is a possible candidate, people say it's very abstract)

  • If one likes to think about mathematical objects from a "geometric" point of view, which book emphasized geometric intuition and connections with other areas of geometry more? (Eisenbud-Harris could be a candidate along with some course on classical algebraic geometry, as the sole purpose of the former is to help a reader to develop an intuition for abstract machinery)

  • Which books is better for a classically-minded reader who is interested mostly in classical algebraic geometry, but wants to understand the modern approach? (possibly, Hartshorne? As he himself says in the preface that he is the classical algebraic geometer, and his books contain a review of classical AG and 2 chapters of applications of the machinery to classical questions, in particular, study of curves and surfaces)

I hope this thread and possible answers could be a useful resource for those starting learning algebraic geometry for the first time.

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  • $\begingroup$ You should specify what "studying scheme theory for the first time" means. Self-study? Reading seminar? Graduate class? (If so, which country, and how long?) Will there be another class or seminar or course in algebraic geometry after this one? All these factors are important. $\endgroup$ – potentially dense Oct 27 '16 at 21:18
  • $\begingroup$ @potentiallydense I'm not asking "which book I should use". It's just a general thread where those who already familiar with some books on scheme theory can shed some light on them for those studying scheme theory for the first time. For instance, I would like some opinions on books mentioned above or other books not mentioned in my post. For example: "I think Vakil's book is good for those... etc.". Or "a categorically-minded reader would certainly benefit from reading...". Something like that. $\endgroup$ – BASIL478 Oct 27 '16 at 21:39
  • $\begingroup$ EGA is the answer to all your problems... it is modern up to this day. The other books are written by people who learned scheme theory from EGA, obviously they cannot surpass the master. $\endgroup$ – syzygy Oct 29 '16 at 12:10
  • $\begingroup$ @syzygy It's quite funny, but I heard the author of EGA (and by coincidence the founder of the field) himself say that EGA is "not modern enough". That is, Grothendieck thought that EGA needs to be rewritten using the modern language of derived categories and abstract (categorical) homological algebra. In particular, he wanted to exclude Cech cohomology in favor of his abstract derived functor definition of sheaf cohomology. So EGA may be indeed the most complete reference on basics of scheme theory, but it surely isn't the most modern one. $\endgroup$ – BASIL478 Oct 31 '16 at 22:12
  • $\begingroup$ Sure, one can say Grothendieck had very "abstract mind" (whatever that means) and the most abstract way isn't the best way to first introduce someone to the material, but I think it would be fair if we at least had one book in accordance with Grothendieck's vision if not out of pedagogical value then at least out of respect for the man. $\endgroup$ – BASIL478 Oct 31 '16 at 22:14

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