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Is this question acceptable?: Is it better to read Hartshorne or EGA to learn modern algebraic geometry?

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    • CommentAuthordavidac897
    • CommentTimeDec 27th 2010
     
    and/or which one is better in which circumstances. This might also be a contrast of different philosophies of doing mathematics. One involves exposition and then lots of problems, the other has no problems, but includes an enormous amount of material, and one can sit down and try to prove as much as possible on one's own.

    This doesn't have one answer, so I would have thought unacceptable, but on the other hand, consider: http://mathoverflow.net/questions/46022/serres-fac-versus-hartshorne-as-an-introduction-to-sheaves-in-algebraic-geometry

    There's also a clear reason why many people would find responses to this question valuable.
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    You might like this question, which asks whether one should ever read EGA.

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    There are a few questions which cover similar ground, so you should try to dig those up first.

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    You might find this helpful.

    Also, I've talked to a bunch of people regarding EGA, and I would be happy to pass on what they told me to you, if you email me (contact info available on my webpage in my MO profile).

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    I can't imagine that anybody would recommend EGA for a first reading in modern algebraic geometry. After you have a feeling for the field then yes, I think it's a great resource; but you'd lose your mind if you tried to learn modern algebraic geometry from scratch with EGA.
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    @Chuck: what do you think people did before Hartshorne was written? :-)
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    @Kevin: What do you think people did before EGA was written? There was only 17 years in between [that is, from the very first volume of EGA to the publication of Hartshorne's book. If you count from the completion of EGA, it's only about 10 years]. A very productive period in the subject of algebraic geometry, but relatively brief in the cosmic scheme of things.

    I would guess that even in, say, 1970, less than half of the people in the world learning "algebraic geometry" were reading EGA. Or do you think that's not the case?

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    @Pete: presumably you mean "before Hartshorne was written"? If not then what I write below is irrelevant. But if you did then, talking to people who were around at that time, my understanding was that they read EGA (more than one person has said to me "you're lucky kid, you don't remember the time before Hartshorne when to learn alg geom you had to read EGA or get your hands on a copy of Mumford's notes", for example). So yeah, at least in the UK, I think that if you wanted to learn alg geom pre-Hartshorne then you typically had to read EGA, or Mumford's Red Book. Do you think different? Admittedly I only have two data points.
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    I think Pete meant EGA.

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    Yes, I meant EGA. Part of what I meant is that although, in some circles, algebraic geometry in the 60's and 70's meant locally ringed spaces and schemes, I'm guessing that nevertheless a lot of people away from Harvard, Princeton, Chicago, Paris etc. were still thinking and learning about algebraic geometry in a more classical way.

    I wasn't there to see it myself, even less than Kevin was (he was born in the late 60's, me in the mid 70's), which is why I keep using the word "guess". My evidence is that I have come across papers in or near algebraic geometry by fantastic mathematicians of the period which are just not written with any acknowledgment of the EGA revolution. The two examples that come most immediately to mind are Shimura and Lang, each of whom wrote for decades afterwards without really adopting the scheme-theoretic language. (Don't be too quick to say that they were not algebraic geometers. I think they definitely were, just not of the Grothendieck-Mumford school.) If such luminaries could get away without using EGA-style geometry, I'm thinking that many lesser -- and less fashionable -- students and researchers did the same.

    P.S.: Mumford's Red Book is a far cry from EGA. For instance, as a graduate student I read through it fairly carefully without any enormous expenditure of time.

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    Oh you did mean EGA: in that case the answer is that they read Weil's foundations. Certainly that's what Swinnerton-Dyer did and surely Shimura et al did that too.
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    @Kevin: sure, I know that. My point was....actually, I forget what my point was. Come to think of it, I believe I have a referee report to write...

    • CommentAuthorBCnrd
    • CommentTimeDec 29th 2010
     
    Dear Pete: The question is about how to learn the theory of schemes, not how people (like Lang & Shimura) who already mastered Weil made the adjustment (if any). My data points agree with Kevin's, in the sense of supporting the impression that those who were trying to learn about schemes in those days (which includes Lang but not Shimura) did face a difficult task due to a paucity of accessible references. Also, Lang actually made the switch (as did Murre) in terms of learning Grothendieck's theory, he had extensive contact with people in the few major centers of the field at the time (which was a much "smaller" field than it is today, as Mumford has indicated in his writing), and he did relearn it by reading in EGA. (I have seen his typed notes on EGA III_1 section 5, so he wasn't just dabbling at the beginning.) Lang's review of EGA for the AMS Bulletin supports your comment about there being plenty to do for those who didn't want to adopt the "new" approach, but the final sentence of Lang's review is instructive in this respect.
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    @B: Thanks for those remarks.

    I think some of my comments have been misconstrued a bit, perhaps because they are peripheral to the topic at hand. (In other words, I both agree with just about everything you said and also agree that your remarks are more on point than what I was saying.) I was indicating that a lot of algebraic geometry throughout the 60s and 70s -- even by superstars like Shimura, Lang, Neron, and so forth -- was not couched in the scheme-theoretic language at all. So I am skeptical about the idea that there was ever an entire generation of students who learned algebraic geometry by reading EGA, at least if you do not construe "algebraic geometry" too narrowly. (Think of Griffiths and Harris, written at about the same time as Hartshorne and not long after the completion of EGA: this is another enormous algebraic geometry book which is remarkably close to being completely disjoint from EGA.) Obviously if you want to restrict to "EGA-style algebraic geometry" then EGA becomes a canonical text.

    If anyone cares about my opinion on the subject (and I'm not sure that you should): as a wannabe arithmetic geometer, I often did not find Hartshorne to be speaking my language. Given all the trouble that one takes to set up the edifice of scheme-theoretic algebraic geometry [here at UGA we cover about one chapter of Hartshorne per semester course, so one course on classical geometry, one course on schemes, one on sheaf cohomology and so forth], to dismiss the study of schemes over non-algebraically closed fields as being somehow too complicated seems very weird to me. My favorite books on the subject are Mumford's Red Book and Qing Liu's relatively new text, which I think has a chance to become the Hartshorne-for-arithmetic-geometers of the future. I have nothing against EGA whatsoever: I just happen not to have read it. But I'm not averse to it nor would I warn anyone else off: as a graduate student I found myself having to learn some material out of SGAVII, and I found it to be thoroughly solid and reliable. If I ever needed to know something and someone told me to look it up in EGA, I would do so happily.

    • CommentAuthorBCnrd
    • CommentTimeDec 29th 2010
     
    Dear Pete: To return to the topic of davidac987's question, I agree with you that Qing Liu's book seems to be extremely well-suited for students of arithmetic geometry, and I encourage davidac987 to take a serious look at it as he is learning the material. In particular, the book's rich supply of examples that involve arithmetic properties of the ground field or base dvr is quite nice, to say nothing of the extensive exercises and self-contained development of the semistable reduction theorem for curves and other important applications. On the flip side, Hartshorne introduces the use of derived functors (which Liu's book avoids), so each book has its own merits. I see it as akin to the task of learning class field theory: better to read around in several introductory accounts than to focus all of one's attention on a single resource. I also agree with Chuck Hague.
    • CommentAuthorAndrewL
    • CommentTimeDec 29th 2010
     

    @Pete I think in the coming years,the whole point will be moot as more and more graduate students will be using Ravi Vakil's online text-especially as he revises it and gets more and more of the bugs out of it. I plan on using it to learn AG in depth after I use Fulton's book and Artin's lecture notes online as background.

    @Kevin "@Chuck: what do you think people did before Hartshorne was written? :-) " If they were smart,they studied topology instead,Kevin.........LOL

    • CommentAuthorEmerton
    • CommentTimeDec 29th 2010
     

    Just to throw in another point of view: I think that the best thing is to learn a little bit of foundations, say from the Red Book, from Hartshorne Chapter I and by dabbling in Chapters II, IV, and V, and then start reading the papers that are of interest to you. Let the rest of your foundational reading be dictated by your research interests and the problems that you are trying to solve. In particular, I wouldn't expend a lot of time trying to learn cohomology of coherent sheaves until you've discovered a coherent cohomology group that you really need to compute. (In any event, the formalism of cohomology is fairly easy to learn, and I think it helps to learn this, and to see the manipulations being made in various places and to become familiar with them, before going carefully into the foundations.)

    • CommentAuthorShevek
    • CommentTimeJan 1st 2011
     
    When I was learning scheme theory I found it very useful to look at scheme theory from a historical perspective and I spent a lot of time (maybe too much) trying to really pin down exactly how the foundations of algebraic geometry developed throughout the 20th century; the state of things in Weil's day; the lead up to schemes; and the aftermath... In particular, be sure to check out Serre's notion of algebraic variety as found in FAC.

    One random reference that I kind of like is the following survey
    I.V. Dolgachev -- Abstract algebraic geometry
    (originally published in Russian in 1972).

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