# The importance of EGA and SGA for “students of today”

That fact that EGA and SGA have played mayor roles is uncontroversial. But they contain many volumes/chapters and going through them would take a lot of time, especially if you do not speak French.

This raises the question if a student, like me, should even bother reading EGA. 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.

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I know this question was aimed with algebraic geometry students in mind, but there are researchers in other subjects which make heavy use of algebraic geometry and I would be interested in comments on whether close knowledge SGA/EGA would be worth the investment for such people as well. –  Tyler Lawson Oct 28 '09 at 13:52
maybe it can be useful to say/link what are EGA et SGA –  Gil Kalai Nov 17 '09 at 22:17

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.)

EGA and SGA are dangerous, because they are so powerful, and thus so tempting. 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.)

• If you can’t use “non-Noetherian rings” in a sentence, you can’t do algebraic geometry. (This is just as silly as saying that if you don’t know the analytic proofs underlying classical Hodge theory, then you can’t call yourself an algebraic geometer. It depends on what you are working on.)

Your goal is to (eventually) prove theorems. You want to get to “the front” as quickly as possible. 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.

Don't forget: EGA and SGA were written at the dawn of a new age. 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. Just because it is in EGA or SGA doens’t mean it is important. Just because it is in EGA and SGA and not elsewhere doesn’t mean it is not important. How will you know the difference?

So when should you read EGA or SGA?

• 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. Most people who read EGA and SGA as students are not in this group.

• Some will read them later as they need facts, and will realize how beautiful they are.

• Some will read them later for "pleasure", like reading the classics.

In summary, you should read EGA and SGA only when you need to, where "need" can have many different meanings.

In fairness, I should say how much of EGA and SGA I've read:

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.

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Some will read them later as they need facts, and will realize how beautiful they are. That is so true. I needed some facts concerning differential operators for my PhD thesis. After asking several experts in related fields and obtaining no clear answers I gave up asking and decided to lift through EGA (or SGA, I've forgot). And there it all was: very clearly and thoroughly stated. I was ashamed I let the immensity of the work scare me away before. If I had had the courage to go to it instead of learning from Hartshorne's alone, I would have been much more happy. –  Anonymous Feb 8 '10 at 22:44
Dear Anonymous, No need to be ashamed --- I've been there too (as have many others). You only know what you are looking for once you are actually looking for it. Perhaps students should be encouraged to dip into EGA a little, to realize that it isn't scary, just long, so when they really need it, they won't be fearful. Also, I wonder if Johan de Jong's stacks project will replace one of EGA/SGA's functions, providing a searchable browsable repository of complete proofs of useful facts. –  Ravi Vakil Feb 9 '10 at 10:11
I think the stacks project is close to approaching this, and indeed may have already approached it and started moving beyond! It is an incredible resource, in terms of its completeness, its level of detail, its generality, its open discussion of its goals, assumptions, and choices of proof, and for many other reasons. And it has the mesmerizing quality of reading EGA, or of a well cross-referenced encyclopedia: you turn to it just to look up one small fact, which leads to another, and another, ... . –  Emerton Feb 9 '10 at 21:21
Matt, I fully agree! –  Ravi Vakil Feb 14 '10 at 22:28
Dear Ravi, your sentence Some will read them later for "pleasure", like reading the classics is very unexpected, profound and thought-provoking. –  Georges Elencwajg Jul 21 '11 at 17:33
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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 … .

This is part of the reason why I advise against spending too much time just holed up with EGA and SGA. 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.

Siu's methods are analytic; one might summarize them as $L^2$-methods. 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.

It's not clear to me that one gains more essential background by reading EGA.

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.)

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Dear Matthew, may I ask (I am just very curious, since you know so much about Grothendick style AG and use it so fluently): how did you learn all that? Would you share with us? I think it will be helpful for many people, including me. –  Hailong Dao Feb 8 '10 at 17:54
Dear Hailong, I learned Grothendieckian algebraic geometry by reading papers of Deligne, Gross, Katz, Mazur, Mumford, and Ribet, and also some of Grothendieck himself (including FGA and some SGA, although not EGA (!), which I have only skimmed a little of). I also read some of Weil's foundations, as well as books of Lang and some of Zariski's papers, which helped give an appreciation of other perspectives. –  Emerton Feb 8 '10 at 18:34
Thanks for that answer. –  Hailong Dao Feb 8 '10 at 21:58

I guess I am a professional algebraic geometer of a kind, albeit not a "non-arithmetic" one as specified in your question. When I was in graduate school I told Joe Harris I wanted to take a year off so I could read EGA and really learn algebraic geometry properly. 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.

In any event, the French, as others have said, is not an obstacle.

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Virtually every page I've read of EGA/SGA has been useful to me, and almost every page I've skimmed I've later wished I'd read in more detail. The reputation for difficulty is, I think, unfounded. They are certainly abstract, but virtually every detail is present; in many ways, that makes EGA/SGA easier to read than other sources. 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.

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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:

Bloch: You can’t tell a student now to go to EGA and learn algebraic geometry...

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. Of course, EGA or SGA are more like dictionaries than books you could read from A to Z.

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: No... You’re exaggerating.

Bloch: You never knew whether behind the veiled curtain was something very interesting that you should search back in a different volume to find; or whether in fact it was just referring to something that was completely obvious and you didn’t need to...

Illusie: That was one principle of Grothendieck: every assertion should be justified, either by a reference or by a proof. Even a “trivial” one. He hated such phrases as “It’s easy to see,” “It’s easily checked.” When he was writing EGA, you see, he was in unknown territory. Though he had a clear general picture, it was easy to go astray. That’s partly why he wanted a justification for everything. He also wanted Dieudonné to be able to understand!

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Dear Fitzcarraldo, The transcript of this conversation is wonderful. But note that Bloch is also an illustrious algebraic geometer, so it is worth noting his opinion as well! Best wishes, Matthew –  Emerton Feb 3 '11 at 19:38
The snarky remark about Dieudonné at the end (and the exclamation mark) are in very poor taste. Dieudonné was a great mathematician and the EGA would not exist without him. More importantly, he is no longer with us to defend himself against this very unfair jab. –  Georges Elencwajg Jul 21 '11 at 18:13
@Georges: Do you think he would've found it offensive? Perhaps it was just a joke, as Dieudonné was more of an analyst than an algebraic geometer. –  Jacob Bell Feb 13 at 10:45

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.

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I agree! I don't know of any references except for Liu's book that are more readable than EGA/SGA (Milne's Etale cohomology covers some SGA ground though). Any suggestions? As for the non-noetherian/finitely presentation stuff...I haven't needed it yet, but the day might come. And then I'll be glad to have EGA/SGA :) –  Lars Oct 28 '09 at 17:58

Regarding EGA, I think the most appropriate answer is: "wy bother ?". Unless you have a really special interest, you shouldn't.

Edit

Expanding on this (it seems a lot of people seems it's just flame): When you are starting to learn algebraic geometry you have to cover a lot of material. Here is a list of what you must cover:

• Hartshorne,
• Some complex-point-of-view book (e.g. Griffith and Harris or Gunning's books)
• Something from an arithmetic point of view (e.g. Silverman)
• Fulton's intersection theory.
• Some book about curves (my preference is ACGH, but there are many options)
• some book about surfaces (e.g. Beauville or but Barth et-al)

All this come before and during reading material in your speciality, anything modern, or anything which is just plain fun. So what are you to do ? Read the list above and some more from your-speciality/modern/fun, or are you going to do alg-geom all over again, but this time with non-noetherian rings ?

end of Edit

SGA s a different beast. I read parts of 4.5, and was glad I did.

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"EGA is written in full generality. Indeed, such generality, that I don't know of any reasonable geometric situation where you need all this generality." Downvoted because there are plenty of reasonable geometric situations where you need more generality than, say, Hartshorne's book or any other given book. Of course I don't think one should initally learn from EGA but I think it deserves more respect than `why bother?'. –  David Zureick-Brown Oct 28 '09 at 19:16
+1 even though I'm not qualified to discuss the thread topic, because I think -2 just for a forthright dissenting view is overly harsh - especially if the opinion is honest. Otherwise, why bother having a discussion on this kind of thread? one could just let the cognoscenti agree –  Captain Oates Oct 29 '09 at 2:00
@Greg: Sorry, I misunderstood your original remark. However, I have a hard time with this point too: Consider early 20 century point set topology: You have more separation axioms then Dewey numbers. You then prove that a combination of some plus insert-favorite-property-here is equivalent to another combination. But why do you do it if there are so few interesting combinations of these axioms ? (now I'll be downvoted again for trashing point set topology). –  David Lehavi Nov 4 '09 at 5:57
You need to consider non-Noetherian schemes when doing some natural constructions in arithmetic geometry. If you're working over a complete discrete valuation ring A then sometimes you might temporarily need the residue field to be alg closed. So you go up to a bigger DVR B by making some base extension. And then you need to go back down to A again by making a descent. But B tensor B over A isn't Noetherian. Deligne once looked at a paper of mine and said "you'd better not assume Noetherian in this lemma or there will be problems later". That's proof enough for me. –  Kevin Buzzard Nov 4 '09 at 23:20
I dislike Hartshorne's book immensely. His constructions are "morally wrong" but "technically correct" (see his construction of the structure sheaf or the sheafification). The only reason to read Hartshorne is for the exercises. –  Harry Gindi Feb 8 '10 at 21:12

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. Etc.

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.

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I am by no means an expert, but I've found both of these useful. Much of Hartshorne is EGA-light (and by light I mean extra light): for example, there are exercises in Hartshorne that are sections in EGA (e.g. affine morphisms (exercise II.5.17 of Hartshorne is EGAII section 1)). EGA can be a hard read, but it is also more complete than Hartshorne.

As for SGA, perhaps there are other places to get this information (I'm no expert), but I've certainly used it for material on reductive groups over arbitrary bases (SGA3 III), etale pi_1 (SGA1), and etale cohomology (SGA4.5).

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I'm not a full-time algebraic geometer, but I have found both EGA and SGA very useful in my research. 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.g., Weil's writing. Once you learn a few standard words, like "soient" and "dans", it's reasonably smooth sailing.

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I am not a 'professional algebraic geometer', but know several mathematicians who use algebraic geometry and my impression is that one really notices the difference between those who worked through significant parts of EGA/SGA and those who didn't. 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.

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My advisor is a specialist in Grothendieck algebraic geometry and Noncommutative algebraic geometry. I ever asked him about reading EGA and SGA. He strongly suggested to finish all the exercises in Hartshorne but not read Hartshorne. And he said it was good to take a look at EGA when you need something.

For SGA, fortunately, it is not necessary for us students to read these materials because he gave us lecture course on his own work which heavily depends on Grothendieck Machine(especially Tohoku lecture,SGA1 SGA 4 SGA6 SGA 4.5 and other book, say J.Giraud's cohomologie non abelienn, Gabriel-Zisman). In fact, we can learn most part of these materials from his lecture course and even more categorical(since he is framework maker for NCAG which is based on category), It is really a happy experience However, he said it was better to listen not to read.

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If you browse MO pages, you'll notice there are many references to SGA and EGA, recommended by professionals :)

Though reading it fully in itself might be not the best idea — there are lots of great introductions to many concepts of algebraic geometry.

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