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    • CommentAuthorRavi Vakil
    • CommentTimeJun 15th 2010 edited
    I want to ask a question *What should be learned in a first serious schemes course?*, and I want to phrase it to be maximally useful (to everyone as well as myself). Anton pointed out that my previous draft was sufficiently undirected that it might not attract concrete useful answers. He suggested I ask for question-crafting advice on meta. (This is my first posting here, so I apologize if I picked the wrong category.) My draft is below. I'm pondering tagging it [big-list] (and [ag.algebraic-geometry]) but not [community-wiki]. My hope is that people will give a few thought-provoking answers, and the zeitgeist may catch some particularly useful ideas and propel them to the top.

    Draft question:

    I've just finished teaching a year-long "foundations of algebraic geometry" class [link ].  It was my third time teaching it, and my notes are gradually converging.  I've enjoyed it for a number of reasons (most of all the students, who were smart, hard-working, and from a variety of fields).  I've particularly  enjoyed talking with experts (some in nearby fields, many active on mathoverflow) about what one should (or must!) do in a first schemes course. I've been pleasantly surprised those who have actually thought about teaching such a course (and hence who know how little can be covered) tend to agree on what is important, even if they are in very different parts of the subject (number theory; arithmetic geometry; that part of topology using algebraic geometry; commutative algebra; classical algebraic geometry; Gromov-Witten theory; etc.).

    *** What topics/examples/ideas etc. really really should be learned in a year-long first serious course in schemes? ***

    Constraints: The hypothetical first course in the question should be purely algebraic. The course should be intended for people in all parts of algebraic geometry. It should attract smart people in nearby areas.  It should not get people as quickly as possible into your particular area of research.   It can (and unfortunately must) be hard. Anything essential must be proved, with no handwaving (e.g. "with a little more work, one can show that...", or using exercises which are unreasonably hard).  Intuition must be given when possible. (Why I'm asking:   I will likely edit the notes further, and hope to post them in chunks over the 2010-11 academic year to provoke further debate.  Some hastily-written thoughts are here[link ], if you are curious.)

    As usual in big-list questions: one topic per answer please. There is little point giving obvious answers (e.g. "definition of a scheme"), so please propose things you think others might forget or disagree with, or things often omitted, or things you wish someone had told you when you were younger. Or propose things people often discuss that you think could be safely omitted. Please motivate your answers.
    • CommentAuthorEmerton
    • CommentTimeJun 15th 2010

    Dear Ravi,

    I think this an interesting question, and I would enjoy reading the answers. (As a caveat: I am far to the left on MO with regards to being liberal in my views about what constitutes an acceptable question!)

    Best wishes,


    I'd be very interested in reading the answers to this question as a student who's been trying for quite some time to learn the subject!
    (Caveat: I'm also rather to the left.)
    I think it's a great question! But I'm also pretty left-wing...

    It might be a good idea to include a list of minimal prerequisites for such a course. But I am in favor of it, and I consider myself pretty centrist.

    So I think the goals in framing these sorts of big list questions should be to try to promote serious answers which make an argument over drive-by answers where people just say the name of a topic and leave. Questions that attract lots of short poor answers are frustrating.

    In this case, one option would be to list the topics that you already cover and tell people that their answers should explain why their topic is worth *displacing* one of the topics that you already have in the class. This will clarify people's thoughts in terms of how important a topic should be in order to be mentioned and thereby cut down on the number of marginal suggestions.

    +1 Noah, good idea. Doing that will certainly make the answers more focused and less driveby-ish.

    • CommentAuthorBoyarsky
    • CommentTimeJun 15th 2010 edited
    Why mention the constraint that everything essential must be proved (in lecture, I assume)? One can easily imagine students benefitting from exposure to some impressive topics developed in an inspired way with selected black boxes to be filled in by the interested student later in life (e.g., if you want to illustrate the utility of spectral sequences in theorems on coherent cohomology, you wouldn't insist on grinding out the monstrosity of the general construction at the blackboard; and likewise if doing derived functors one can explain the underlying principles and use the theory to great effect without having had to develop the foundations at the blackboard. And likewise for the analytification functor -- fine to say the course is largely algebraic, but insisting on omission of any discussion of analytic topics sounds too restrictive -- discussion of specific moduli spaces, etc.)

    It is fair to request that suggested topics should be able to be developed to the point of communicating deep/important ideas and/or techniques whose utility can be illustrated with examples. But insisting that everything essential must be proved for any topic discussed in the course runs into practical problems, since the intro aspects of this field simply cannot be shoehorned into a year-long course and so there has to be some slack in the system somewhere: either coverage of fewer topics in greater detail, or more topics with some (but not all) details omitted (with emphasis on including examples). Each viewpoint has its own merits.
    • CommentAuthorRavi Vakil
    • CommentTimeJun 15th 2010
    Noah, I potentially like your idea, but I fear that I might shape the answers by giving a template which can be edited, rather than potentially starting from scratch and making some radical choices. (I experimented with some radical choices, and some I stuck with, and others I didn't. I tried doing with derived functors, but settled for using them judiciously. I tried doing quasicoherent sheaves in detail before even defining morphisms of schemes, but the participants wisely rebelled --- they had a good sense of how to think categorically. I won't say what radical choices I stuck with for fear of steering the discussion in a certain direction.) I feel like many algebraic geometry courses currently taught are of this sort: take a standard source, and add and subtract, but don't alter the overarching storyline. I don't mean to knock this --- I think most people (myself included) teach most classes this way --- it takes too much effort to rethink every course from scratch. But by giving an embedded link, I hope to give people the chance to click through, and perhaps to be outraged that I included something that should reasonably be cut, or excluded something essential.

    Boyarsky, you propose an excellent different model for such a course. (And there are others too --- for example, a Griffiths&Harris course, or a course designed around elliptic curves, or many more.) By constraining the question, I hope to constrain the answers, and avoid larger and potentially pointless debates about what kind of course is "best". (I'm quite catholic about this --- we intend to have analytic courses alternating with algebraic courses, with arithmetic courses always happening at the same time.) Of course, I fully expect a number of responders to bring alternate models up, and possibly there will be overwhelming support for one of them --- or perhaps that a first course should not restrict itself to being only one kind of course.

    There were several reasons why my initial iteration of this course (as a mass reading course) involved not skipping details. I've found that as a teacher, I was tempted to be lax in skipping details, and I realized later that students really couldn't fill in details themselves --- I would of course omit the hardest ones, and those would be precisely the ones that students wouldn't be equipped to do. (My real test was: could students do exercises? If not, the fault was mine, not theirs. Would they all avoid certain exercises in particular?) In fact I was such a student (and many of my peers were) --- only gradually later did I learn how to fill in details. And your examples are good ones. I agree with your philosophy on spectral sequences: I explained how to use them, and gave lots of exercises that were as easy as possible, and then gave a short written proof of why they worked, and told them not to read it until years later when they felt they had to, and then they should just read it once and then burn it. But I encouraged them to skim through it and to realize that they could understand it if they wanted to. Derived functors I claim can actually be done at the blackboard, so long as you explain them the right way (with examples, and no black boxes, and no derived categories), and participants do exercises. The restriction on avoiding analytic issues (except as remarks and intuition) was just because already such a course was too full, and adding more topics would require subtracting others. In short, the two issues are related: it cost some time being complete, but far less time than I'd initially feared. But the main trade-off which made this work was having well-defined boundaries of the scope of the course (no forays into Sobolev spaces etc.). But being scrupulous in this regard made clear which of the topics covered were hard, and which topics were not, because you could see all the moving parts.

    (And finally, I feel that if participants can't solve problems and do exercises, they don't know the material. But this opinion isn't universal --- for many people, a first course is just an exposure to the subject, and they learn how to prove things later, on their own.)

    So perhaps I should reword the question slightly: there are a few key results that I think one can leave out (I usually prefer to given a short written proof), but they should be as few as possible. (For me, examples include Krull's Principal Ideal Theorem, the local criterion for flatness, and the guts of spectral sequences.)
    • CommentAuthorCSiegel
    • CommentTimeJun 16th 2010
    I like the question, and am looking forward to a chance to put in my answers, from the point of view of someone who's taken two "first" schemes courses, one that went well, and the other that...didn't.

    I would make this question community wiki. Saying "one topic per post" implies that you wish to use the voting system as a popularity contest for the topics. This is not quite its current role since, via the link to reputation, its current role is to encourage helpful and useful answers and, by extension, discourage answers that are likely to be voted against. It seems as though you wish to encourage quite a broad range of answers and not discourage answers that may be voted against.

    I realise that you want some defence of the topics, but I would be surprised if you got much detail there. I think that MO works best when:

    1. The questioner feels that they could answer the question themselves if they spent a lot of time on it, but only really wants the answer and not (really) the process by which the answer is obtained
    2. The answerer can answer the question very quickly by dint of knowledge that the questioner doesn't have

    The point is that when asking a question, one is asking someone else to do something for you. As there are no real direct incentives, the best way to get someone to do this is by making it easy for them. Saying "Please motivate your answers" is asking for details, reasoned arguments, engagement in discussion, and so forth. Whilst you may get a few answers that provide all of this, it may scare off others from answering at all.

    I would be surprised if there was a topic that could be taught in this course that you couldn't figure out the motivation! So I would focus it more on the key time-saver: ideas that have worked in a first course on ... what was it? ... schemes. You can then follow-up any interesting ideas, either briefly in comments or more fully via email. Indeed, if at first you ask for ideas and then in a comment on one you think good you ask for more details then I think that you are more likely to get the details than if you ask for them upfront - my reasoning being that by specifically asking for details on a specific answer, you are indicating to the answerer that you are particularly interested in their answer so they will realise that their answer is helpful directly to you and so be more inclined to provide the details.

    Finally, it may just be the time-of-the-year, but questions that have phrases like "Please motivate your answers." make me think of exams and - completely illogically - I find myself not answering such questions even if I know an answer. (Not that I would have an answer for this question!)

    To sum up:

    1. Make it community wiki, but more importantly:
    2. Figure out how to maximise your benefit from the answers whilst minimising the effort that someone has to go to to provide a useful answer.
    3. Good luck with the question!
    • CommentAuthorRavi Vakil
    • CommentTimeJun 16th 2010
    Everyone: your comments are very useful!

    A current draft is at
    I fear it is getting a little long for a quick question.

    Following Noah's advice: I've added some focus, to try to get people to give responses that involve taking a stand (last paragraph).
    I don't think I mind "drive-by" answers: if some pithy one-liner catches people's fancy, that would be interesting to see.

    Following Boyarsky's advice, I've made clearer that the constraints are not because this is a royal road into the subject, but just one of many possible paths. Hopefully people will also respond if they disagree strongly that such a course should exist. This is not an unreasonable opinion: "Students should get a sense of the big picture, and not worry about particular details."

    Andrew, thanks for many helpful points. I'm happy with this being community-wiki. (I've informally heard arguments both ways. Any further opinions on community-wiki out there? Pros and cons?) I've edited the "Please motivate your answers" a bit --- I hope this better focuses things so people give some justification, but please feel free to let me know if more should be done. I don't fear people not answering. With schemes in particular, people tend to have opinions (having had, as CSiegel did, good and not-so-good experiences). I think this is for historical reasons (and I could digress on this). One key sentence you wrote: "I would be surprised if there was a topic that could be taught in this course that you couldn't figure out the motivation!" Again, with schemes in particular (for historical and cultural reasons), I don't think this is true: there are topics that people do (or do early) just because everyone does them, without asking the motivation. Forcing people to say why a topic should be included opens up some interesting discussions. (Example, at risk of starting a discussion inappropriate for meta: "Of course you have to do valuative criteria to prove properties of separated and proper morphisms!")
    • CommentAuthorRavi Vakil
    • CommentTimeJun 16th 2010
    I almost forgot: in response to Qiaochu, prerequisites are now embedded in the question as an option.

    Very quick response on a couple of things:

    opens up some interesting discussions

    and that might be the most frustrating part of the whole thing! Discussions just don't work on MO. My suggestions were based around the fact that, from experience, the way to get the most out of a question on MO is to make it so that there isn't much back-and-forth but just a question-and-answer.

    One key sentence you wrote: "I would be surprised if there was a topic that could be taught in this course that you couldn't figure out the motivation!"

    If I'd written my sentence in Norwegian it might have been clearer that the "you" that I was referring to was very definitely Ravi Vakil. I'm sure that there are lots of things taught about schemes that Ola Nordmann couldn't motivate, but I'll be amazed if there's a good answer given that Ravi Vakil couldn't motivate!

    Of course some people will give details and motivation and that's to be encouraged, but one has to be careful about asking for it up front since the way that it is done may dissuade people from contributing at all.

    Typos and other more specific comments:

    1. "I've been pleasantly surprised that" -> "I've been pleasantly surprised to find that"
    2. "Certainly most excellent first courses that ignore ..." -> "Certainly most excellent first courses ignore ..."
    3. The last two constraints (hard and rigorous) seems a little orthogonal to the question. Whilst I'm sure that there are some topics that couldn't be done at that level in a rigorous way, the majority will have some variability in how hard or rigorous they are done (note that I'm speaking with no experience of the subject whatsoever!). So insisting on these constraints confuses me a little as to exactly what is wanted, and would make me hesitant at proffering an answer (not that I could, of course, I'm trying to imagine what it's like to be an algebraic geometer and failing miserably; actually, just failing - I'm quite happy with not being an algebraic geometer).
    4. I'd take the "Why I'm asking" paragraph out of parentheses. Given that the first sentence says that you've just finished this course, it's not absolutely clear that this isn't a "What should I have done differently?" rather than a "What should I do differently next time?". That paragraph is therefore useful in setting the scene for the scheme.
    5. Thinking more about the last paragraph and all the pleading, I'd turn them into, effectively, a description of the most useful answer. For example, "so please propose things" could be "I'm particularly interested in things". Make it more personal!
    6. Lastly, where's the question? I'm half-joking, but it's not signalled in words, only by formatting. A simple "Thus my question is:" at the end of the first paragraph would make it much more obvious.

    Dear Ravi: I do hope that you'll put your notes on the internet when you teach this class, since it seems like you're putting a lot of work into thinking it all out.


    +1 Harry. I agree, I've been reading your old notes and I think they are great! I would love to see what the new notes look like, once they are ready.

    • CommentAuthorRavi Vakil
    • CommentTimeJun 16th 2010
    I'm running to a meeting and will write more later, but the draft is edited in partial response to Andrew's helpful comments. A longer response will come later today!

    Now that I read the first paragraph with my suggestion in place, it doesn't work. You go straight from talking with experts to asking the question and I'm left wondering why you need to ask this question given that you've had such great conversations with other experts. So I recommend that the link sentence be longer, and make it clearer that one doesn't have to be an expert to be able to answer this - indeed, I would expect some of your most useful answers coming from those who've just taken that course and found that certain things "just clicked" whilst others went way over their heads. Something like: "Useful as that has been, I'm worried that there's things that I've overlooked. So I'd like to hear from a wider community. Thus my question is ...". Or something! I'm sure you know someone better qualified than me to iron out the wrinkles in the English!

    (Just to make one thing clear: as originally written, it is far clearer than the majority of questions on MO. But since Ravi asked for comments, I'm giving my opinions. I quite like the idea of polishing a question a little before it gets asked on MO particularly when, as in this case, the question itself is not quite a usual MO question.)

    • CommentAuthorRavi Vakil
    • CommentTimeJun 16th 2010
    Andrew, I've re-patched. I'm now trying to be careful not to always make it longer with each edit, so I've contracted it slightly as well. There hasn't been any further support for community-wiki, so my default option is not to do it (as that can be changed later if the audience wants it). About discussions on MO: I've often found the give-and-take in the comments to be quite enlightening, and that's what I'm hoping for here.

    Harry and Gretar, I'm intending to gradually put the notes online over the next year (even before I teach it again). (Gretar, thanks for the kind words.) The reason I am not putting the current version online is that it is rougher near the end, and there are numerous things that need to be done. By putting the notes online at a steady pace, I hope to force myself to edit them at a steady pace. I also hope that a few brave souls will treat them as a world-wide online reading course, and that experts will check in periodically to see how I present various topics, and then weigh in with improvements.

    The comments here seem to be converging, so I'll likely post this question Thursday (June 17) morning (PST) unless further comments suggest otherwise.

    When you say "putting the notes online at a steady pace" do you mean every week, or less frequently? The reason i ask is that I'd love to go through the entire set of notes, and i think it would keep me motivated to do it roughly at the same time as you go through the course at Stanford.

    • CommentAuthorRavi Vakil
    • CommentTimeJun 16th 2010
    Gretar, I'm not going to teach the class here next year, but I was thinking of posting notes at the pace of a course. I was thinking of doing it every second week. I wouldn't want to do it less frequently. Every week is possible too. If that would help keep people motivated, that would be pretty convincing to me. I might do something lasting longer than an academic year, as the notes are intended to be a big course plus a little more, so people can skip sections depending on their predilections.

    A back-of-an-envelope calculation: say September through June, averaging 25 pages every 2 weeks. (Many pages would not be part of the "course", but just available as "side reading". But that pace is still very fast.) But people's responses on MO may change my plans!
    Ravi: In line with Gretar's question, you might consider opening an RSS feed that indicates updates to your notes. (I wish more people did this.)
    • CommentAuthorRavi Vakil
    • CommentTimeJun 16th 2010
    Tyler: I'll likely distribute them in a wordpress blog. I think that gives the option for viewers to get an RSS feed. Is there another reasonable option?
    Ravi: A blog would certainly work, hopefully be pretty straightforward for you, and you get the feed by default.

    Outside that, a feed is roughly as difficult to make by hand as a standard HTML page, or I believe there are some utilities that will do it for you.

    Dear Ravi, this is a bit more to ask than just posting lecture notes, but would you at least consider filming the lectures and putting them up on the internet as well? I've never seen an online lecture series on AG/Schemes before (and I've been told that the introductory AG class this year at Michigan is going to be very focused on varieities. If not, then thank you anyway for graciously agreeing to put your notes online =).

    Possible typo in your web page "it is very helpful to learn our algebraic geometry at the same time as your commutative algebra." The implications of this phrase are intriguing, but I suspect that you mean to write "your" both times.

    Notes every other week sounds fantastic, and doing it as a blog is also great. I look forward to September.

    • CommentAuthorRavi Vakil
    • CommentTimeJun 17th 2010
    Harry, I know filming isn't going to happen for a number of reasons. (The first killer is that it costs money. But then there are follow up reasons.)

    David, thanks for catching that! I wish I could say that was intentional.

    Gretar, thanks --- I'll begin to set this up soon, and see whether people feel strongly about going to once per week.
    • CommentAuthorCSiegel
    • CommentTimeJun 17th 2010
    I'll be more than happy with whatever you do, I need to brush up on my scheme theory anyway. Will you post here about where you'll be posting the notes? You'd mentioned wordpress.
    • CommentAuthorRavi Vakil
    • CommentTimeJun 21st 2010
    I've now set up where I'll post the notes: . I intend to start around September 1, in order to give myself some time away from the notes. And as people here already know, the mathoverflow question is here: . Suggestions of how best to do this (e.g. perhaps every week rather than every 2 weeks) are welcome (here, by email, or at
    Thanking you for doing all this. I am curious if we might be able to incorporate things like this into Math-Online.For example, listing things that might follow (or precede) your FOAG course. In this way we would end up with some sort of curriculum.