Timeline for The importance of EGA and SGA for "students of today"
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
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| Apr 11, 2019 at 12:42 | comment | added | user137767 | @KevinBuzzard slightly prophetic comment I would say, now that the world has seen perfectoids. | |
| Feb 8, 2010 at 21:12 | comment | added | Harry Gindi | 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. | |
| Nov 4, 2009 at 23:20 | comment | added | Kevin Buzzard | 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. | |
| Nov 4, 2009 at 5:57 | comment | added | David Lehavi | @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). | |
| Nov 3, 2009 at 22:12 | comment | added | Greg Stevenson | @David: I did not at all mean that one can simply wish away noetherian hypotheses! I meant it in the technical sense that writing things as colimits of noetherian rings one can sometimes reduce to the noetherian case and prove what you want there. One example of this is Thomason's proof of tensor nilpotence (in fact this seems unavoidable to get the proof in the right generality I have a different proof of his theorem about thick subcategories which also needs this reduction). But it is true that not everyone cares about these things. | |
| Nov 3, 2009 at 14:12 | comment | added | David Lehavi | @Taylor: my original answer and the discussion following it are really getting out of hand... I did not say "don't ever read EGA"; all I said was "unless you have a really special interest". It seems you do. I still maintain that almost all practicing algebraic geometers has never needed - or will need - the non-noetherian case, and that this is what EGA so long and hard to read. | |
| Nov 3, 2009 at 13:36 | comment | added | Tyler Lawson | +1 for the more detailed explanation of your reasons. But the Lazard ring representing formal group laws is a polynomial ring in infinitely many variables, and the moduli of formal groups has a correspondingly non-Noetherian structure. That's one example, but of course not everybody is interested in such things. | |
| Nov 3, 2009 at 11:57 | comment | added | David Lehavi | @Greg: you can't simply remove the noetherian hypothesis from theorems and get away with it. The thing is, I don't know of any (and it seems that neither do any of the people who downvoted this opinion) interesting example where noetherianity does not hold. | |
| Nov 3, 2009 at 11:03 | comment | added | Greg Stevenson | It can occur that non-noetherian critters can have more interesting structure/pathologies which in some cases are very informative. I think if nothing else it is good to see removal of noetherian hypotheses at some stage just so one knows it can be done and to get a sense of what the natural hypotheses on schemes are for certain things. | |
| Nov 3, 2009 at 6:30 | history | edited | David Lehavi | CC BY-SA 2.5 |
Explaining why it's not flame
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| Nov 3, 2009 at 6:10 | comment | added | David Lehavi | @Anton: ok - making it a little more informed. I never encounter an interesting algbro-geometric situation where the underlying ring was not Noetherian. The chief reason EGA is such an unpleasant read is that he deals with this case. | |
| Nov 3, 2009 at 3:54 | comment | added | Anton Geraschenko | -1 @Yemon: Voting on an opinion question indicates (dis)agreement. I strongly disagree with this answer, so I'm voting it down. If the answer was informed and dissenting, I would probably still vote it down to show my disagreement, but this answer is uninformed (or at the very least, uninformative). It's as though someone said to you "unless you have some really specialized application in mind, why bother studying non-abelian groups?" | |
| Oct 29, 2009 at 2:00 | comment | added | Yemon Choi | +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 | |
| Oct 29, 2009 at 0:18 | history | made wiki | Post Made Community Wiki by David Zureick-Brown | ||
| Oct 28, 2009 at 19:16 | comment | added | David Zureick-Brown | "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?'. | |
| Oct 28, 2009 at 17:18 | comment | added | David Lehavi | - whoever downvoted this, it's common curtsy to leave a remark when you downvote | |
| Oct 28, 2009 at 17:18 | comment | added | David Lehavi | @Ho: 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. | |
| Oct 28, 2009 at 15:56 | comment | added | user709 | Can you please elaborate on the reason for not going into EGA? Too technical? Hartshorne good enough? or..? | |
| Oct 28, 2009 at 15:32 | history | answered | David Lehavi | CC BY-SA 2.5 |