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in reply to Jim's question over on the blog, I was looking for a free spot on the nLab where I could write some general motivating remarks on the point of "derived geometry".
I then noticed that the entry higher geometry had been effectively empty. So I wrote there now an "Idea"-section and then another section specifically devoted to the idea of derived geometry.
(@Zoran: in similar previous cases we used to have a quarrel afterwards on to which extent Lurie's perspective incorporates or not other people's approaches. I tied to comment on that and make it clear as far as I understand it, but please feel free to add more of a different point of view.)
edited and expanded higher geometry - contents a bit
I tied to comment on that and make it clear as far as I understand it
I do not understand then...I mean you are now explicitly that it is questionable weather the DERIVED noncommutative algebraic geometry of Kontsevich fit with commutative derived geometry of Lurie, which I do not object on, and I did support it with their standard saying that the derived world does not see the difference between commutative and noncommutative. However it should say DERIVED. The nonderived noncommutative alg geometry based on abelian categories does not fit yet, to my knowledge (it is far more difficult than the derived geometry in any of the two senses, the one based on truncated objects like derived and the one based on coherent infinity picture). Finally there is an old definition, often useful, that one looks just at derived category and not enhanced one as an invariant, this is different as I claim on derived algebraic geometry page; this is manifestly not the same as Simpson-Toen-Lurie. But due to recent theorem of Lunts and Orlov the importqance of that is not big. Namely, for quasiprojective algebraic varieties there is a unique dg-enhacement, so in practice both framework should lead to the same results for good variety-like objects.
I should say, that there is a point of view difference: one can always consider the usual scheme as a derived scheme in two ways. One is non-forgetful not loosing any information. Another is to REPLACE scheme by (enhanced version or not) of the derived category of coherent sheaves on it. This is what Russians do since 1977-1978 Beilinson, and 1988-1989 Kapranov. This is of course easier geometry, some information of a scheme is lost. That does not mean that it can not fit into Simpson-Toen-Lurie approach. It can, but the word derived geometry implies we never look at usual schemes with full information. What is lost is finite information for variety case. Namely having some simple choices like ample line bundle may often fix a situation (e.g. there is such a theorem of Bondal and Orlov).
one can always consider the usual scheme as a derived scheme in two ways. [...] Another is to REPLACE scheme by (enhanced version or not) of the derived category of coherent sheaves on it.
Yes. And I would think both points of view fit nicely in the, as you say, Simpson-Toen-Lurie-approach -- but combined with stabilization.
Here is what I mean:
in that approach, too, given a scheme, we may think of it just as the oo-stack on simplicial rings^op that it represents.
OR, we may turn it into a "structured oo-topos" by considering the oo-topos of oo-stacks over the scheme. The stabilization of this oo-topos (or rather its homotopy category) should be something like the derived category of the scheme.
Right, but unlike Lurie I do not like to say derived scheme for higher enhancement of usual. I like to say derived scheme for either derived category or (edited: typo) enhanced derived category of usual. Toen is more careful, he has saz algebraic geometry based on model categories in one series of works and then takes many special cases like brave new geometry, derived geometry etc.
Lurie in a sense does not cover strictly the cases which some other approaches do, as in positive characteristics the A infinity world is rather different from stable category world, but this is the matter of choice of sister frameworks, rfather than philosophical.
While for generic terms of higher geometry I can accept your choice to look at them as two expressions for the same thing, i still like distinguishing talking derived SCHEME versus higher scheme following Toen-Vezzosi-Vaquie distinction which is very meaningful: the faithful embedding of the 1-category of usual schemes into higher schemes Aff op -> sSet preserves limits, while into derived higher schemes DAff op -> sSet does not; this distinction makes the distinction between the terminology important and more general than the two examples in the question. One can of course consider still the limits from the point of view of the 1-subcategory of schemes internally.
If one looks at nonfaithful image of usual schemes (you use word stabilization here), then one has the third possibility for limits, within the stable world. I however suspect that for good examples this additional difference is not essential. I do not know enough on say geometric stacks to argue however.
I agree, Zoran, that's a good point.
Maybe we can eventually find a good system of terminology for talking about all things derived geometric. People have created quite a terminological mess.
I agree, I was thinking in the bus about changing some of my own preferences and I think it is improtant that we influence that carefully. Thus we should try to sort out (I am also thinking of redoing those parts of those entries which I done earlier before some of my attitudes changed) this things, but maybe not too quickly. It is not easy to account for the sutbleties and reasons of the particular subfields motivating the terminology, and still fit with the general picture (which is, even worse, beyond infinity categories i daresay).
Archiving an obsolete discussion from derived algebraic geometry
There is no really systematic rule to the use of the word “derived” here. For instance derived stack has become the standard term for the general version of the notion of ∞-stack, but Higher Topos Theory is not called “derived topos theory”.
Urs Schreiber: personally I’d think that “derived algebraic geometry” is therefore a misnomer. But who am I to stop that train? :-)
Zoran: this paragraph is entirely wrong, hence your repenting it. There are two generalizations needed to come from schemes to algebraic geometry: deriving on the left and deriving on the right. The deriving on the left corresponds to take higher algebraic stacks,say in terms of fibrant objects in certain model category of simplicial presheaves. The deriving on the left means taking the fibre products of schemes in certain derived way as well (amounting to taking the left derived functors of the tensor product on the algebra level), but the model structures here take the flxibility of dg algebras in the source of the simplicial presheaf picture; this takes care of nontransersality. Thus derived stack is not only higher stack, it is also derived on the other side.
Hey Zoran, why did you leave a rude comment like that on my MO post and vote it down? I wasn’t talking about dimension theory, I was talking about being “n-geometric for finite n” and that an “n-geometric non-derived algebraic stack is n+1-truncated”. That wasn’t right.
I think I left a polite comment. Somebody gave you -1 as it should be as the answer is totally incomprehensible, somewhere in the deep waters of something you just talk to yourself. I think it is impolite to press on others how they will grade your answer so I added -1 partly for your impoliteness, partly for your quoting in hi-brow manner very difficult and specific theorems not to do with the question. The issue is very simple. It was polite to explain why I also voted -1 as the unknown guy before me. Finally you did not construct any example, what the guy asked for.
@Zoran: Part of the question was regarding $\infty$-stacks (non-truncated), so I cited the theorem that shows that in the non-derived case, all higher algebraic stacks are truncated.
I also gave a reference to where the OP could find an example.
Regarding the original -1, on the contrary, it is not polite to vote an answer or question down without leaving a comment.
Also, some people over at MO just vote down any question or answer I submit, regardless of the quality, and if someone leaves a comment like yours, people will just vote it up (and my post down) because they think I’m a jerk and they like to see me shamed. If you just want to comment on the quality of my answer, that’s fine, but if you want to criticize my behavior or whatever, I would really really really appreciate it if you would contact me either by e-mail or on the nforum or something.
No hard feelings though.
=)
Hm. Who erased my comment though ?
Regarding the original -1, on the contrary, it is not polite to vote an answer or question down without leaving a comment.
I disagree, I think both commented and uncommented vote is equally polite. I know when I am a referee that a low quality aspect of something can much easier be detected than explained and that to do the second step is often the unsuccessful waste of time.
and if someone leaves a comment like yours, people will just vote it up
I know, hypocritical political correctness is quite in fashion, especially in US. It is utterly antiintelectual.
but if you want to criticize my behavior or whatever
Who criticise your behaviour ? You must talk about somebody else. I have only written that I agree with the voter, that I disagree with you asking why he voted (this should be without pressures) and that the article is incomprehensible/difficult for a simple matter.
Now your article disappeared as well. Not only my comment. It is getting weird.
By the way I left one simple comment to the main question there at some point. It is just a general explanation, but I did not try to propose an answer as I am not very good with examples in this field (I know them pretty superficially), what was kind of emphasised in the question.
Because I deleted it.
There’s nothing hard or incomprehensible about it.
Also, regarding political correctness or whatever, I was asking you to do me a favor, not to censor yourself.
I disagree, I think both commented and uncommented vote is equally polite. I know when I am a referee that a low quality aspect of something can much easier be detected than explained and that to do the second step is often the unsuccessful waste of time.
Right, because I wouldn’t understand a two line comment explaining why my five line post got voted down… It would be a real waste of the person’s breath. I don’t know if it’s what you meant, but it seems like you’re saying that I’m not even worth a minute of your (or whoever else’s) time.
By the way I left one simple comment to the main question there at some point. It is just a general explanation, but I did not try to propose an answer as I am not very good with examples in this field (I know them pretty superficially), what was kind of emphasised in the question.
Yeah, and I told the OP to watch Toen’s talk on nonabelian hodge theory for an in-depth explanation of a specific example.
Anyway, the level of respect you have for the average voter on MO is unwarranted. For instance, I asked a question, and somebody posted an answer that was complete nonsense. It received three votes up (more than my question!) before I called it out as being nonsense (Andrew found out that the poster in question did this to several posts and received a number of votes up before the answers were deleted as spam).
I think it’s completely fair to ask why a certain vote was made, given the track record of the average MO reader.
I don’t know if it’s what you meant, but it seems like you’re saying that I’m not even worth a minute of your (or whoever else’s) time.
On the contrary, while in the MO discussion I did take the stand of going into (though minimal) discussion, in this entry above I defend the right of the opposite stand, that it is a usual habit that such discussions are not worthy, and even more unsuccesful, the latter obviously true with you as well. The guy who asks the question does not know about the basic motivation, and you still consider the subtlety on dimensions of categorical constructions involved in some particular approach by Toen et al. as “nothing hard”. While I understand that you are now on top of these things, most of us are not, and hence we find it difficult to get from chapter 6, to paraphrase Chomsky.
I think it’s completely fair to ask why a certain vote was made, given the track record of the average MO reader.
Well, I prefer to communicate with MO community not looking at its greatest common denominator. But I can understand where you are coming from. But even then voting down even by low sophisticated visitor is, I believe, either if something is incomprehensible or incorrect; in the latter people are more likely to directly react with a comment, what is logical. More rarely there are some other aspects like bringing nonmathematical issues into the answer what is clearly not the case in your post. Hence I think that you could have been reasonably sure that the vote was about the accessibility of your answer.
Back to some discussions with Urs: survey of a book (this one: pdf)
http://www.tau.ac.il/~corry/publications/reviews/pdf/kromer.pdf
says
A first main focus of attention in Krömer’s book concerns the evolution of three mathematical disciplines in their interaction with CT between the mid-1940s and the late 1950s: algebraic topology, homological algebra and algebraic geometry. As Krömer rightly points out, in spite of their names, these three disciplines are quite different in nature, and his narrative shows an illuminating interconnection among them with CT as a common axis. Indeed, Krömer shows in a detailed fashion how tools developed in each of them were successfully applied in the others.
I think that Urs and I both disagree with that opinion. I mean it is not the matter of good device of interconnection, really the three subjects in their foundations are quite closely related and parts of a whole. Of course, there are more specific and specialistic questions in each of them, but this is not a point.
Zoran, thanks for discussing this with me. I appreciate it! I see your point now. Thanks.
+1 Harry and Zoran for sorting this out quickly. :)
I think this entire voting system is broken, is a shame, and would better be abandoned. But of course it won’t. So it should be ignored as much as possible.
Right, it is not a good system, but drove more people into the MathOverflow activity than without it. In long term it does not make sense. Urs, do you agree about he Kroemer issue above ?
Urs, do you agree about he Kroemer issue above ?
Haven’t looked at the article, but the piece that you have quoted sounds quite reasonable.
Including “these three disciplines are quite different in nature”? I think they are almost indistinguishable in their essence and wholeness.
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