The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series. The material has a reputation of being hard to read for a number of reasons. The style is very abstract and makes heavy use of category theory. Moreover, an attempt was made to achieve maximally general statements, while assuming that the reader is aware of the motivations and concrete examples. These were published as the seminar proceeded, beginning in the early 60's and continuing through most of the decade.
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There must nowadays be less time-consuming ways to absorb the "required knowledge" needed to do "serious" non-arithmetic algebraic geometry. I would like to hear what professional algebraic geometers would recommend their students in this matter. At the time I write this, there are a number of wise words already written here, so I'll add just incremental thoughts.
Much of what I might say was already said by Matt Emerton. It is easy to be mesmerized. Their writing is roughly synonymous with the founding of modern algebraic geometry as a field.
But I think their presence has driven people away from algebraic geometry, because they give a misleading impression of the flavor of the subject. And even writing "flavor" in the singular is silly: there are now many vastly different cuisines.
But this is less true than a generation ago. Here are some well-known negative consequences. Relatively few successful practicing researchers hold these views, but I have heard them expressed more than once by younger people. There is a common feeling that there is an overwhelming amount one has to know just to understand the literature.
To be clear: one has to know a lot, and there is even something that can be reasonably called a "canon" that will apply to many people. But I think the fundamentals of the field are more broad and shallow than narrow and deep. It is also true that much of the literature is not written in a reader-friendly way.
Those who can quote chapter and verse of EGA are the best suited to doing algebraic geometry. To be clear: some of the best can do this. But this isn't a cause of them being able to do the kind of work they do; it is an effect.
It depends on what you are working on. Your goal is to eventually prove theorems. You want to be able to do exercises, then answer questions, then ask questions, then do something new. You may think that you need to know everything in order to move forward, but this is not true. Learn what you need, do some reading for fun, and do no more. The rules were being written, and these were never intended to be final drafts; witness the constant revisions to EGA, as the authors keep going back to improve what came before.
These ideas have been digested ever since. How will you know the difference? A very small minority can and should and will read them as students. But you have to be thinking about certain kind of problems, and your mind must work in a certain way. A small part of EGA I've read in detail.
I had a great time in a "seminar of pain" with a number of other people who were also already reasonably happy with Hartshorne and more. Reading the first two books of EGA with some guidance from Brian Conrad on what to skip was quite an experience I had assumed that it would be like Hartshorne, only more so, with huge heavy machinery constantly being dropped on my head.
Instead, each statement was small and trivial, yet they inexorably added up to something incredibly powerful. Grothendieck's metaphor of opening a walnut by soaking it in water is remarkably apt.
But other than that, I've read sections here and there. I'm very very happy with what I've read I agree with Jonathan on this , and I'm also happy knowing I can read more when I need to, without feeling any need to read any more right now I have better ways to spend my time.
When I need to know where something is, I just ask someone. And as for my students: I'd say a third of my students have a good facility with EGA and possibly parts of SGA, and the rest wouldn't have looked at them; it depends on what they think about.
I'm surprised, reading the various answers and comments to this question, how much support there is for the idea of reading EGA. It is evidently a mathematical masterpiece of a certain kind, but I would never recommend it to a student to study.
In response to a similar question asked on Terry Tao's blog, I posted the following advice :. As to how much time to spend on EGA and SGA, this is something that you ultimately have to decide for yourself, hopefully with the guidance of your thesis adviser.
But one thing to remember is that many very clever people have pored over the details of EGA and SGA for many years now, and it so it is going to be hard for anyone to find interesting new results that can be obtained just by applying the ideas from these sources alone as important as those ideas are.
Even if you want to make progress in a very general, abstract setting, you will need ideas to come from somewhere, motivated perhaps by some new phenomenon you observe in geometry, or number theory, or arithmetic, or …. By themselves, they are not likely for most people to provide the inspiration for new results. On the other hand, when you are trying to prove your theorems, you might well find techincal tools in them which are very helpful, so it is useful to have some sense of what is in them and what sort of tools they provide.
But you will likely have to find your inspiration elsewhere. I think it is worth thinking about two of the most significant recent theorems in algebraic geometry: the deformation invariance of plurigenera for varieties of general type, proved by Siu, and the finite generation of the canoncial ring for varieties of general type, proved by Birkar, Cascini, Hacon, and McKernan, and independently by Siu.
I'm not sure what text one would begin with to learn these methods. Certainly Griffiths and Harris for the very basics, but then The methods of BCHM are techniques of projective and birational geometry. A careful reading of Hartshorne, especially the last two chapters, would be a good preparation for entering the research literature in this subject, I think.
An aside: there is much more to EGA than just handling non-Noetherian schemes, but the spectre of non-Noetherian situations seems to loom a little large over this discussion. Thus it seems worthwhile to mention that, while non-Noetherian schemes arise naturally in certain contexts as Kevin Buzzard noted in a comment on David Levahi's answer , I think these contexts are pretty uncommon unless one is doing a certain style of arithmetic geometry.
Somewhat more generally, I don't think that flat descent should be the focus for most students when learning algebraic geometry. It is a basic foundational technique, but I don't expect most interesting new work in algebraic geometry to occur in the foundations.
More generally still, this is probably a good summary for my case against spending time reading EGA. I guess I am a professional algebraic geometer of a kind, albeit not a "non-arithmetic" one as specified in your question. He told me not to be absurd, that I should just learn techniques as I needed them for problems. I don't take a position as extreme as that.
It's true that I haven't in any real sense "read" EGA. On the other hand, I have it on my shelf and am perfectly comfortable referring to it when necessary.
I do think the project of reading EGA would be of great value to an aspiring algebraic geometer; but there are lots of other projects with this property, and I don't think reading EGA is indispensable in a way these other projects are not.
I suppose I end up giving a rather mushy answer; if reading EGA appeals to you, then you will probably be drawn to the kind of problems where it's essential that you've read EGA. If not, then maybe not. The reputation for difficulty is, I think, unfounded. Opening a volume and reading a sub-paragraph from the middle can be difficult because of all the back-references, but reading linearly can be very pleasant and rewarding. The French language may be a barrier for some, but one doesn't have to "learn French" to learn enough to understand EGA.
I just would like to draw your attention to what Luc Illusie, illustrious student of Grothendieck, said to Spencer Bloch in a conversation recently published in the Notices of the AMS:. Illusie: Actually, students want to read EGA.
They understand that for specific questions they have to go to this place, the only place where they can find a satisfactory answer. You have to give them the key to enter there, explain to them the basic language. And then they usually prefer EGA to other expository books. Bloch: One thing that always drove me crazy about EGA was the excessive back referencing. I mean there would be a sentence and then a seven-digit number Illusie: That was one principle of Grothendieck: every assertion should be justified, either by a reference or by a proof.
Though he had a clear general picture, it was easy to go astray. I think a student should try to read as much as possible from EGA and SGA, especially if he is interested in arithmetic geometry.
Hartshorne's book is good, but at times he considers only schemes over algebraically closed fields, where he could be more general. I mostly use them for seeking out specific theorems and constructions, but I also do occasional light browsing for "culture". For people who tend toward my sort of use-case, it is probably good to be sufficiently familiar with these works to recognize questions that might be answered in them, but not necessarily able to quote chapter and verse.
I don't think I have the attention span or the time nowadays to read them front-to-back. Edit: I should say something about the French. The language in EGA and SGA uses a very restricted vocabulary and simple sentence structure, so you don't run into the sort of elaborate turns of phrase you'd find in e.
Once you learn a few standard words, like "soient" and "dans", it's reasonably smooth sailing. Further, Grothendieck wrote EGA for learning algebraic geometry, one needs only little french to read it and in any case learning french is a big personal extension. I think EGAs and SGAs are not useful for "students of today" but they are indispensable for "researchers of today", and "tomorrow".
There is just so much stuff there that is not available anywhere else. I think it is common when learning algebraic geometry to return to the same subjects over and over again but at a different level.
First you skim the subjects to have a general idea and to orient yourself. Then you start working on a concrete research problem and you learn the tools to deal with it and your immediate algebro-geometric neighborhood. Then you revisit the same topics and, with your new and better perspective, you appreciate more of it. So EGAs are perhaps for the visit 3, or 4. And the more you visit, the more pleasant it is to see things done elegantly and in full as humanly and humanely possible generality.
I am by no means an expert, but I've found both of these useful. EGA can be a hard read, but it is also more complete than Hartshorne.
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