There seems to be no activity on this site lately. So, unless someone objects convincingly, this site will die on january 15th. Perhaps you may consider joining MathOverflow? all the best for 2010 :: lieven.
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lieven lebruyn
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lieven lebruyn
SIGMA calls for papers
An email from the editors of SIGMA :
“We would like to announce a special issue of the journal SIGMA (Symmetry, Integrability and Geometry: Methods and applications) “Noncommutative Spaces and Fields”.
Deadline for paper submission is March 30, 2010.
There is no limit to the length of an article. Both original research articles and review papers will be considered (not having been published elsewhere).
Publication will be done only on the results of the peer review according to refereeing policy of SIGMA.
More information is available at this website
You may forward this information to anyone you know who might be interested in the subject.”
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lieven lebruyn
Liu & Yau hit the arXiv (again)
Today, Chien-Hao Liu and Shing-Tung Yau (yep, the one of Calabi-Yau fame) posted a paper titled Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a B-field background à la Polchinski-Grothendieck Ansatz. (Btw. is there a limit to the number of characters one can put in an arXiv-title? Surely, it should be less than a text-message?)
To get my attention, they might as well have stopped after the first comma : ‘Nontrivial Azumaya noncommutative schemes’. If there is one noncommutative scheme which isn’t noncommutative at all, it should be an Azumaya scheme surely … nontrivial, or not.
But hey, I downloaded the pdf-file and surely enough their Key-words included all of the current arty-farty-fanshi-wanshi terms : Azumaya scheme, Azumaya structure, B-field, D-brane, D-string, D-module, deformation quantization, gerbe, Higgs/spectral pair, moduli stack, morphism, Polchinski-Grothendieck Ansatz, quantum spectral curve, sheaf of algebras of differential operators, twisted sheaf, valuation criterion.
Wow! Perhaps, I should at least try to understand some of this…
Woops … I must have been living in another galaxy or have been preoccupied with entirely different things lately, for this is but the 5th (!) in a series of papers with equally catchy titles. Here’s the list:
- (part 1) Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski’s D-branes in string theory from Grothendieck’s viewpoint
- (part 2) Morphisms from Azumaya prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curves
- (part 3) Azumaya structure on D-branes and resolution of ADE orbifold singularities revisited: Douglas-Moore vs. Polchinski-Grothendieck
- (part 4) Azumaya structure on D-branes and deformations and resolutions of a conifold revisited: Klebanov-Strassler-Witten vs. Polchinski-Grothendieck
Yeah well, my bad. So, I’ve printed it all out, made a pot of coffee and started reading…
I’m finished for now, but don’t quite get it. So, has anyone worked through these papers and got something out of them?
In part 1 they define ‘Azumaya-type noncommutative space’ which appear to be just an
-sheaf of orders on a (commutative) scheme 
. But, for their ‘physical’ applications they seem to require that this order is a split Azumaya algebra (in parts 1-4) and a general sheaf of Azumaya algebras (in part 5). Here’s what I believe to understand of their approach. I’ll just sketch the affine case (that is, working with rings rather than schemes) as they use only tame (that is, central) localizations to glue things together.
They want to have a ‘geometric’ (ie. space and stuff) interpretation of the natural extension of Grothendieck’s ‘functor of points’ approach. That is, they consider for a given algebra
the covariant functor 
and want to construct their ‘geometric’ object containing as much information as this functor. In the cmmutative case, Grothendieck showed how one could do this : take the prime spectrum of the algebra and adorn it with a structure sheaf, that is, look at the affine scheme
of your algebra. Then he showed that their is a duality between the two approaches, that is, to any algebra morphism 
, there is a corresponding scheme-morphism 
.In the noncommutative case we face the immediate problem that there isn’t even a map from prime ideals of
to prime ideals of !Lie & Yau solve this problem as follows : their ‘geometric’ object associated to a complex algebra
does not just consists of twosided prime ideals of itself, but also of all prime ideals of all (!) 
-subrings of . Might sound like a crazy idea at first, but it ties in nicely with old results by Artin and Procesi.They showed that there is(!) a natural map between prime ideals whenever the algebra map
is a ‘central extension’, meaning that 
with 
being the center of .
So, what Liu&Yau propose it to take all prime ideals of all possible central extensions to resolve the problem.For example, an Azumaya-type noncommutative point of size n, corresponds to
should really be thought of as all prime-ideals of all 
-subalgebras of . They call it the ‘noncommutative cloud’. As a consequence, their ‘generalized morphisms’ from this noncommutative point to one of their “Azumaya-type noncommutative spaces” associated to an algebra is nothing but the collection of all n-dimensional semi-simple representations of .Right, now that we connect their approach with the little we algebraists know, would any of you be interested in running through these papers together in a couple of posts, trying to connect the NCAG-content to ’stringy’ stuff like D-branes, Chan-Paton modules and Higgsing/un-Higgsing, whathever all this means?
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lieven lebruyn
the NAG canon
This is a follow-up on my nLab-rant before. Of course, John and Urs have more than a point that I should put in some work if I feel some of the information contained in their nLab is not up-to-par. So, here’s my first attempt to canonize some of the Noncommutative Algebraic Geometry papers :
Noncommutative algebraic geometry (NAG) extends algebraic geometric methods (local rings, intersection theory etc.) to study noncommutative algebras, and conversely, uses noncommutative algebras in the study of commutative algebraic varieties (Brauer groups, noncommutative desingularizations, stacks etc.).
Arguably, the first genuine NAG-paper was Mike Artin’s “On Azumaya algebras and finite dimensional representations of rings” (J. Alg. 1969), but there are precursors which turned out to be influential in later developments : the two Auslander-Goldman papers “Maximal orders” and “The Brauer group of a commutative ring”, both in Trans. AMS 1960, and, Pierre Gabriel’s foundational paper “Des categories Abeliennes” (Bull. Soc. Math. France 1962).
Artin’s 1969 paper characterizes Azumaya algebras via their identities and paved the way to generalize algebraic geometry to rings satisfying polynomial identities, via their representations and invariant theory. Subsequent major contributions were made by Claudio Procesi with his book “Rings with polynomial identities” (Marcel Dekker 1973) and his paper “The invariant theory of n x n matrices” (Adv. Math. 1976) and by Mike Artin and Bill Schelter’s “A version of Zariski’s main theorem for PI rings” (Amer. J. Math. 1979). The link with representation theory, using seminal work by Paul Cohn and George Bergman, was clarified by Claus Ringel “The simple Artinian spectrum of a finite dimensional algebra” (in Dekker Lect. Notes 51, 1979) and culminated in Aidan Schofield’s book “Representations of rings over skew fields” (1985). These developments have been influencial for recent work in ‘formal’ noncommutative geometry, such as Maxim Kontsevich “Formal (non)commutative symplectic geometry” (1993), Joachim Cuntz and Daniel Quillen’s “Algebra extensions and nonsingularity” (JAMS 1995), “Noncommutative smooth spaces” by Kontsevich and Alex Rosenberg (arXiv 1998).
In the 70ties, localization theory was popular among ringtheorists, and, several people tried to generalize structure sheaves to noncommutative algebras. Gabriel-like localizations via kernel functors led to the books by Jonathan Golan “Localization of noncommutative rings” (Marcel Dekker 1975) and Fred Van Oystaeyen “Prime spectra in noncommutative algebra” (Springer LNM 444, 1975), and making the link with the Artin-Procesi approach, the book by Fred Van Oystaeyen and Alain Verschoren “Non-commutative algebraic geometry” (Springer LNM 887, 1981). Universal localizations were used by Paul Cohn in “The affine scheme of a general ring” (in Springer LNM 753, 1979) and later by Claus Ringel and Aidan Schofield as mentioned before. In England, Goldie’s localization theory led to the theory of ‘cliques’ and ‘clans’ of prime ideals (work by Jategaonkar, Hajarnavis and others) which may in retrospect be viewed as influential to present local description via quivers and A-infinity structures.
In the early 80ties maximal orders were revisited as noncommutative versions of normal varieties. In particular their local structure was investigated. Extending prior work by Mark Ramras, Mike Artin succeeded in describing the etale local structure of maximal orders over surfaces. In trying to generalize this to higher dimensions, one was quickly led to impose strong homological properties on the orders (and later, more general algebras). This led to the investigation of homological homogeneous rings (work by Ken Brown and others), Artin-Schelter regular algebras and, finally, Auslander-regular rings.
A huge project started, trying to classify Auslander-regular algebras, graded-connected and generated in degree one (that is, noncommutative analogons of projective n-1-space) for small dimensions n. In dimension 3, the initial work was done by Artin and Schelter, culminating in the influential papers ATV1 and ATV2 (ATV for Mike Artin, John Tate and Michel Van den Bergh, ATV1 in Grothendieck Festschrift 1990, ATV2 in Invent. Math 1991). In this classification a class of algebras re-appeared which were initially discovered by Odeskii and Feigin, the so called Sklyanin algebras. Ringtheoretic investigations of the Sklyanin algebras were done by Tate and Van den Bergh (Noetherianity for all dimensions n) and in dimension 4 a geometric study via ‘point’ and ‘line’ modules was carried out by Thierry levasseur, Toby Stafford and Paul Smith. Further classifications in dimension 3 (dropping the degree one generator condition) and 4 were done by these people and their students. In the ungraded case, maximal orders with excellent homological properties were recently used in connection with quotient-singularities and their desingularizations as in the noncommutative crepant resolution by Michel Van den Bergh (Abel symposium 2002).
The use of categories in NAG has increased in recent years. First, in the definition of the proj of a graded ring as the quotient category of all graded modules by the finite length modules. Next, in the ‘derived’ geometry, originally developed by Bondal and Orlov, and generalized to the noncommutative case by Bondal and Van den Bergh in “Generators and representability of functors in commutative and noncommutative geometry” Moscow MJ 2003, and then, in deformation quantization and related topics.
Right, you can stop scanning now, that’s it. It’s not perfect at all but the best I could do right now on the verge of leaving for London for a couple of days. I’m sure I missed some important stuff. So, feel free to edit the nLab’s noncommutative algebraic geometry page further to your&all’s liking. Urs (of the n-cat cafe) has written a primer to get you started. And, believe me, it really is that simple…
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Eric
I’ve also added a link to this article
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Toby Bartels
Yes indeed! Welcome to the n-Lab.
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Zoran Škoda
Dear Prof. Le Bruyn, I just posted an answer to your previous post (“nlab-rant”) clarifying that entry to Timeline was NOT written by me, and was cut and paste from wikipedia by somebody else and it does not anyway claim that 1998 work is THE entry point of NAG, (it rather correctly says that the idea of representing spaces by categories from Gabriel, Grothendieck, Manin is A entgry point of NAG), neither ideologically nor historically. I did however write the first UNIFINISHED (please appreciate our limitations in time, we risk loosing my job by dedicating part of my worktime not to finish papers needed for my career but doing public service) version fo NAG entry, and I thank you for your kindly adding new material on the ring-theoretic part of NAG. I am sorry that I made you understood the point of the article: it was not at all conceived as a HISTORY of nag (unlike Timeline which is about history of categorical part of mathematics). It is supposed to have modern, very general and categorically biased (ncatlab) exposition of the main ideas of NAG at large. Thus the notions like affine, projective, equivariant NAG, approaches via algebras, graded algebras, dg-algebras, categories, triangulated, abelian or pretriangulated dg-categories should also be mentioned, as well as the spectral methods and localization theory. It would be very nice of you if you coudl further contribute to entreis which are difficult for us to write, and probably easy for you (not only having in mind your knowledge but also because you apparently wrote many manuscripts as an active professor and blogger in the area). For example, entries on maximal orders, Azumaya algebras and Brower group, PI-rings…And please when you do contribute, leave a note at latest+changes page — contributors always leave note there on significant changes to notify other collaborators in our web project.
The deformation quantization part is still missing to balance the nag entry.
Finally the idea of derived geometry started with Kapranov 1985 (influenced by Beilinson 1977-1978) and Kapranov-Bondal in late 1980-s and then was taken by Orlov-Bondal collaboration and Kontsevich. I would like to understand your comment saying that the work on quivers influenced Konstevich et al work on formal noncommutative symplectic geometry. I knwo that some of the main examples come from quivers and deformed preprojective alegbras, but do not understand if you meant some more general role of quivers which i am not aware of.
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lieven lebruyn
Hi Zoran, please call me lieven. John Baez already clarified that the Timeline entry was a copy/paste operation from wikipedia.
ill be happy to add to nLab over time on the few areas i know something about and will try to leave a note on the latest changes page (i was unaware of this). as to deformation quantization andderived geometry i hope somebody else will add/change things as i dont consider myself an expert.
i didnt mean quivers, but rather Cohn’s&Schofield’s work on universal localization, coproducts, hereditary rings, relative cotangent sequences etc. these reappear later in Cuntz-Quillen and Kontsevich-Rosenberg, not always with proper refernces.
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David Ben-Zvi
As Zoran mentions, Beilinson’s 1978 paper on the derived category of projective space (together with the paper it accompanies, by Berstein-Gelfand-Gelfand on Koszul duality) seems like a good place to start an unofficial history of the algebraic geometry of derived categories of coherent sheaves, in particular the idea that derived categories of sheaves are concrete algebraic/geoemtric object (though of course the idea that geometry of any kind is about categories of sheaves goes back to Grothendieck).
One tricky thing though about such histories is that many of the most important papers don’t explicitly herald themselves as studying noncommutative geometry. IMHO one of the seminal results of noncommutative geometry is Beilinson-Bernstein’s 1981 work on localization, identifying categories of representations with categories of D-modules on the flag variety, though maybe it’s not usually considered in that context. And much work on geometric descriptions of derived categories pales in comparison to (or is directly descended from) the epic work of Mike Hopkins with Smith and Devinatz in the ’80s on the nilpotence and periodicity conjectures in stable homotopy theory. These papers show (following a vision of Morava and Ravenel) that the stable homotopy category has a beautiful geometric description in terms of moduli spaces of formal groups. I think algebraic geometers (such as myself) have to a large extent been ignorant of the deep and sophisticated algebraic geometry practiced for decades by algebraic topologists, who have to deal with categories far more subtle than those we typically encounter.
Finally I think it’s hard to overestimate Kontsevich’s influence. Not that I thinking someone here was, just thought it bears repeating.
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Zoran Škoda
I shuld also write that whatever is the Timeline entry on nc schemes pasted from wikipedia it is plainly wrong in the very first sentence which outlines what Rosenberg’s nc scheme is (I should write a separate correct entry on nc schemes). There is no genuine sheaf there for example (in reconstruction one gets a STACK of local categories). Thus whoever wrote may not be even “associated” to me, because I read most of the paper, and would not write this (while a part of the entry is correct). On the other hand, I would like to popularize Tomasz Maszczyk’s work which is achieving similar things in monoidal context. Unfortunately only small part of the Maszcyzk’s work is available online on arXiv, and is about to get systematized as a part of his habilitation thesis in Warszawa. Galois theory, Tannakian theory, various facts about corings, Grothendieck’s Galois theory, Lunts-Rosenberg differential operators on nc rings etc. all are special cases of Maszczyk’s theory (unfortunately the part on the web is not as general as some of his unpublished results). Philosophically Maszcyzk’s work depends on monoidal reconstruction of schemes theorems of Balmer and Garkusha.
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lieven lebruyn
‘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?
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javier
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.
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Eric
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.
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John Baez
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.
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Urs Schreiber
“Seems a strange way to build a knowledge-base, to me.”
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?
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lieven lebruyn
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…
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jim stasheff
Is there a page that says what n-geom means? i.e. only algebraic?
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lievenlb
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.
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Urs Schreiber
Maybe, I’ll follow John’s suggestion above and will add that information one of these days…
Please do.
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Zoran Škoda
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.
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NCAGfan
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.
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NCAGfan
I think the correct way to modify the words”starting point” should be “starting point of general theory of noncommutative algebraic geometry”
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Urs Schreiber
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?
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David Ben-Zvi
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.
NCAGfan and 8 others are discussing. 
lieven lebruyn
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Manny Reyes 4:27 pm on January 4, 2010 Permalink |
I have been periodically visiting this site in hopes that the activity would pick up, so I will be very sad to see it go. Not a “convincing objection,” but a lament.