‘history’ of noncommutative algebraic geometry
The nLab is a great n-category cafe spin-off aimed at “people interested in discussion of expository and research nature about mathematics, physics and philosophy in the light of category theory and higher category theory”.
Usually I land there following up a comment-link at the n-category cafe. This morning, I was reading John Baez’ comment and wanted to know more of the linked nLab’s timeline of category theory and related mathematics.
A good read, even if you are only interested in algebraic geometry. But then I nearly choked on the 1998-Alexander Rosenberg entry. ‘His’ work is hailed as the ’starting point in noncommutative algebraic geometry’. Yeez…
Naturally, I did click through to their page on noncommutative algebraic geometry, which wasn’t a good idea, mood-wise. It appears to be the ‘history’ of the subject, but must be written by someone who wasn’t actively involved back then, or suffers from Alzheimer.
The nLab-about page stipulates : “If you find yourself annoyed by the state any given entry is in, for whatever reason, please feel encouraged to edit it in order to improve the situation. If you feel existing material needs to be changed, you can do so.”
Seems a strange way to build a knowledge-base, to me.
I can understand that you leave a ’stub’ when you don’t know enough of the subject, hoping that someone who does, comes along, and, adds to it. Here the strategy seems to be : fill the page anyway, and, hope that experts are sufficiently provoked by its content to feel the need to rectify.
But then, here we are.
Which important contribution do you think should be added? Or, am I the only one imagining things?
David Ben-Zvi, 
NCAGfan and 8 others are discussing.
javier 3:19 pm on July 19, 2009 Permalink |
I miss a few words about the use of quantum groups as a geometric object on their own, understood as symmetry groups of noncommutative spaces. I think that idea goes back to Manin in the late 80’s.
Don´t have a clue on what you are seeing (or not seeing) to feel provoked, but that´s likely to be a consequence on my own ignorance on the historical developments of the topic.
Eric 7:34 pm on July 20, 2009 Permalink |
The Timeline page you refer to is mostly copied from what was on Wikipedia and is also being populated by someone I don’t know (who I am tempted to consider a “rogue” contributor, since he ignores any suggestions). I wouldn’t condemn the nLab for this weakness regarding the history of noncommutative algebraic geometry. Please PLEASE feel free to correct anything that is incorrect. That goes for anyone reading this comment.
PS: I’m happy David Corfield linked this this blog. It looks interesting. I’m subscribing to the RSS feed.
John Baez 9:33 pm on July 20, 2009 Permalink |
lievenlb: if you want to improve the nLab in some particular way, please do it yourself. It’s like the Wikipedia: everyone does what they can; it’s not perfect, but it gradually gets better. And it’s vastly more efficient for you to click “Edit” and improve what someone wrote, than to write a blog entry complaining and hope that someone will become sufficiently provoked to ask you how to improve the situation, and then do it for you.
Eric: the reason the Timeline page was “mostly copied from what was on Wikipedia” was that Rafael Borowiecki first wrote the Wikipedia article and then, at my urging, put that information on the nLab. It’s not a perfect timeline, but it’s vastly better than no timeline. I put a few hours into formatting it more nicely.
Urs Schreiber 9:55 pm on July 20, 2009 Permalink |
I think a good way to build a knowledge-base is to get people together who have expertise in a given topic and let them try to contribute their knowledge and sort out controversies by civilized discussion where they occur. In lack of the power to force them to do so, the next best idea is to ask them to do so on an About page, trusting that a good scientific community spirit will be at work.
Looking at the history of the page in question, so far we have Zoran Skoda going forward with a contribution of his knowledge and perspective on the topic in question. He is in close contact with Alexander Rosenberg, which may induce a certain angle on his perspective, but even if that perspective is questioned by other experts, as it happens among experts, it hardly undermines the value of this contribution of our generally esteemed contributor.
What we don’t have yet is an idea of which contribution you think should be added.
I am sure in personal face-to-face discussion with Zoran you would now lay out your point of view , then Zoran would have a fair chance to reply, and so on. Likely the two of you would easily agree on various further additional points to add to the entry, that previous contributors didn’t think of or didn’t find the time and energy to provide.. The result of that discussion would likely be of general value and be a likely candidate to be fed into the nLab for the greater benefit of all of us.
So: Which contribution do you think should be added?
lieven lebruyn 1:41 pm on July 21, 2009 Permalink |
Urs, thank you so much for the additional information about the source of that page. i have no problem reading it as representing Alex Rosenberg’s personal recollections. unfortunately, your lesser informed reader has no way of knowing the content is biased. Maybe, I’ll follow John’s suggestion above and will add that information one of these days…
jim stasheff 11:07 pm on July 20, 2009 Permalink |
Is there a page that says what n-geom means? i.e. only algebraic?
lievenlb 11:18 am on July 21, 2009 Permalink |
The FAQ page states “This is a place where people hang out to talk about noncommutative geometry.” A link is given to teh wikipedia page on noncommutative geometry and though this page may not be perfect, it contains both noncomm. DIFFERENTIAL geometry and nonc. ALGEBRAIC geometry. So, one would very much welcome contributions from NDG-people.
Urs Schreiber 8:46 pm on July 21, 2009 Permalink |
Please do.
the NAG canon « the n-geometry cafe 4:46 pm on July 24, 2009 Permalink |
[...] of Abelian Categ…Greg Muller on Flat Families of Abelian Categ…Urs Schreiber on ‘history’ of nonco…paul smith on Introduction to [...]
Zoran Škoda 11:59 am on July 30, 2009 Permalink |
Well Urs and Lieven, please not that I did NOT write the 1998 Rosenberg’s work entry in the Timeline at ncatlab. It existed before I knew what timeline is and was cut and pasted from the wikipedia entry. On the other hand, the sentence in question does not say (it would be ridiculuous) that Rosenberg’s NC schemes and Rosenberg’s work on reconstruction (another paper) are THE startiong point of nc geometry. Instead, it continuous a preceding Timeline sentence saying that spaces can be represented by the A infty, derived or abelian categories of qcoh or coh sheaves. And it says that THIS (more general idea of representing spaces by categories of qcoh sheaves) is A (not THE) entry point of noncommutative algebraic geometry, thus it is much wider and still just a point not the point. Chronologically this statement belongs to Grothendieck-Manin (to study a space you do not need a space but the category of sheaves on this would be space), Gabriel 1960 thesis work (published 1962), Kapranov’s work around 1985 and Bondal’s derived philosophy from around 1989 (the latest according to M.Kontsevich). Thus the entry is correct if understood with correct English.
Having said that I am responsible for some other changes to Timeline and for the first version of entry on NAG, which was very unfinished. We work hard and can not proceed in completing all planned entries, so we first write what is easier and needs less dugging into literature. Thanks to Le Bruyn for adding more material there, though we’d like more explanations, and ideas, rather than only mainly dry survey of bibliography. I agree with well over 95% of what he put there, it represents correct account of a ring-theoretic part of the story. Today NAG is conceived much more generally, for example Sullivan rational homotopy theory extend Spec to dg algebras in a way explained in Getzler’s work and should be considered an early 1970-s work in NAG if understood in most modern way.
NCAGfan 2:39 am on August 3, 2009 Permalink |
as Zoran mentioned, the sentence”A.Rosenberg’s work is the starting point of Noncommutative Algebraic geometry” is not right. I think the philosiphy of NCAG is proposed by Grothendieck(to do geometry, we do not need a real space but category of quasicoherent sheaves on that”WOULD BE SPACE”). Actually maybe Serre was the first guy to adopt this philosiphy(although in commutative projective geometry). Later, Gabriel proved that this philosiphy was right in Noetherian case using his injective spectrum. Around 1980s A.Rosenberg has proved that this philosiphy is right in any case(Although his paper appeared in 1998, Soilbelman told me that Rosenberg knew this result before 1990). I should mention, according to Rosenberg’s paper, Manin used this philosiphy earlier.
Moreover, Artin-Zhang have used this philosiphy in there work”Noncommutative projective scheme(1994)” Therefore, I think the correct starting point of Noncommutative algebraic geometry(emphasis on identifying space with abelian category) is Grothendieck-Manin-Gabriel. However, A.Rosenberg’s work on noncommutative scheme and reconstruction theorem proved that the Grothendieck’s philosiphy is “RIGHT” in any case. Which means that Artin-Zhang-Smith&Michael Van den Berg&Toddy stafford’s work on noncommutative projective scheme can be justified by Rosenberg’s reconstruction theorem. (Which means that “taken any abelian category as a scheme” can be done without worring about anything.) Then,people can built the Grothendieck’s style of framework of noncommutative algebraic geometry(just like what Kontsevich-Rosenberg do,such as flat descent, nc-spaces and nc-stack.) without worring about anything. So in this sense, the “starting point”make sense.
Whatever, I think one should say starting point of NAG is Grothendieck-Gabriel-Manin.
Futhermore, what Zoran mentioned identifying space with derived category of quasi coherent sheaf (or coherent sheaf) may called”derived noncommutative algebraic geometry” it was originated from Kapranov,Beilinson,Bondal-Orlov, Kontsevich-Soibelman…(Moscow school) and Van den Berg.
At last, I think A.Rosenberg’s K theory on right exact categories(in his sense)should be mentioned. He developed his homological algebra on right exact category following Grothendieck’s Tohoku paper and developed Higher algebraic K theory and proved that this K theory is universal.(It is still not clear whether Quillen’s K theory is universal or not).
I think the page in ncatlab describing the history of Noncommutative algebraic geometry is pretty good. Actually, as Professor lebruyn mentioned, The work of Scofield on universal localization should be added. A.Rosenberg’s 1988 stockholm report gave a nice algebraic-geometric interpretation(sheafification) on Gabriel localization in abelian category(in particular, module category). I think this paper is not wide published but it appeared in Kontsevich-Rosenberg’s paper(noncommutative stack)in short form.
NCAGfan 2:54 am on August 3, 2009 Permalink |
I think the correct way to modify the words”starting point” should be “starting point of general theory of noncommutative algebraic geometry”
Urs Schreiber 6:44 pm on September 26, 2009 Permalink |
Today I found the time to do some editorial work on the nLab entry on noncommutative algebraic geometry. Please have a look:
http://ncatlab.org/nlab/show/noncommutative+algebraic+geometry
Some things I did:
added section headlines and a table of contents.
expanded the entry bit
added a few introductory words to the bit on spectra of abelian categories
added lots of hyperlinks, many to existing entries, some to entries that should eventually be created
A general question I have is:
in the “derived noncommutative algebraic geometry” approach one regards A-oo categories and/or triangulated dg-categories as characterizing nc-spaces by thinking of them as cats of quasicoherent sheaves on these would-be spaces.
Now, A-oo cats and triangulated dg-cats are actualy models for stable (oo,1)-cats. These in turn are special (oo,1)-toposes. SO the idea of characterizing a generalized space by an A_oo/triangulated dg-category is exactly analogous (in fact a special case of) the idea of regarding a topos/higher topos as a generalized space.
Has this perspective been made explicit anywhere?
David Ben-Zvi 8:35 pm on September 27, 2009 Permalink |
Urs, The idea that categories of quasicoherent sheaves are analogs of topoi is certainly implicit in noncommutative algebraic geometry, but I don’t know where it’s explicit. However I wouldn’t say that one is a special case of the other, rather that one is a linearization of the other. Namely, rather than assigning to an algebra (or a scheme) the (oo-)category of sheaves of sets (or spaces) over it, such as those represented by other algebras, we are linearizing/stabilizing/Goodwillie-differentiating this assignment and considering sheaves of modules instead. Thus while of course you’re right that stable (oo,1)-categories are special cases of oo-topoi, the usual functor from schemes to the former is not the same as the usual functor to the latter, but rather is its stabilization.
A related point is that in my (highly unoriginal) view, NAG is not about algebras, but about these (oo-)categories of modules (ie algebras up to Morita equivalence) , so for example it seems misleading to me to look for topologies on the category of algebras, rather than on some category or 2-category of categories (say the oo-category of stable presentable oo-categories) in order to formulate NAG (though far smarter people disagree). (In particular I don’t see why non-accessible categories are relevant?) It’s also nice to note that most common categories are derived equivalent to modules over a (derived) algebra, so the difference mentioned in that n-lab page between NC algebraic geometry and NC topology is less pronounced in the derived setting.