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I added a link to a MathOverflow answer which I found helpful in understanding the connection between the two senses of "derived algebraic geometry". I wonder if there's anything more that can be said about this?
Also I'm wondering whether the first sense, i.e. the study of derived categories of coherent sheaves, is really a common use of the term "derived algebraic geometry"; I've always seen it in the second sense.
And perhaps we should mention homotopical algebraic geometry as well?
Also I’m wondering whether the first sense, i.e. the study of derived categories of coherent sheaves, is really a common use of the term “derived algebraic geometry”; I’ve always seen it in the second sense.
What about homological mirror symmetry ? It is about the oldest and most developed style of derived algebraic geometry, with lots of concrete examples. By definition, it is phrased as equivalence of $A_\infty$-categories (of coherent sheaves on B-side).
In general, the distinction is between looking at the $(\infty,1)$-categories/$A_\infty$0categories as OBJECTS of the category of spaces, or we want instead that objects form an infinity-category.
I think most people mean “geometry over a site of formal duals to simplicial rings” or more specifically “over a site of dg-algebras in negative degree” when saying “derived algebraic geometry”. The entry doesn’t reflect that well, I think.
More abstractly, “derived geometry” is geometry in infinity-toposes that are not 1-localic.
Zoran, I haven't heard of homological mirror symmetry being called "derived algebraic geometry". I admit that I know almost nothing about these things, but I was under the impression that this term always refers to the schools of Toën-Vezzosi, Lurie, etc. in which one considers derived schemes or stacks. I guess one could define a derived variety as a (enhanced) triangulated category that can be realized as the derived category of the coherent sheaves on some variety, but I don't think this is the approach of the Bondal-Orlov school, is it? From what little I have read of their papers they view the derived category as basically a convenient package of classically interesting data about a variety (and not as a generalization or extension of the notion of a variety).
My point is that perhaps there should be a separate page about derived categories of coherent sheaves. It seems to me that the relation is basically just a formal one (as you summarized in your last sentence, Zoran).
I entirely agree with what you (Adeel) say here. I think the material currently at derived algebraic geometry is a bit of a mess. The correct definition is hidden there somewhere in a remark under “relation to higher algebra”.
We have an entry derived geometry which describes the general situation. The entry derived algebraic geometry should specialize the discussion there specifically to sites of duals of simplicial rings and of duals of dg-algebras with components in negative degree.
Guys, these are the facets of the same subject.
The derived algebraic geometry of Toen is mainly based on spaces locally represented by COMMUTATIVE dgas. Even for usual noncommutative algebras, not even dgas, the picture with Grothendieck topology is insufficient as it is well known in noncommutative algebraic geometry. The usual schemes are locally represented by commutative algebras which are in the same time generators of their categories of quasicoherent modules. If one passes to the derived category than the appropriate generator in derived sense will be a dga or A-infinity algebra. Interesting enough even for projective schemes one usually has affine situation at the derived level.(hence the whole derived scheme is represented by a single global algebra) the derived categories of coherent or qcoh sheaves or of perfect complexes do fit with the dga view of derived geometry; the only thing is that for noncommutative dga’s one has less developed theory (of gluing etc.) on schematic side. Kontsevich has replaced the object of study by general $A_\infty$-algebra what allows noncommutative examples; you can view it as a $k$-linear stable $(\infty,1)$-category; not quite a $(\infty,1)$-topos, but similar. It is a generalization, for example (the derived categories of) Landau-Ginzburg models are among the new examples there. The description of the category of Chow motives is much simpler in this wider world. If one localizes at Lefschetz motive, one can embed the localized category of Chow motives into the Kontsevich’s noncommutative motives which are rather simpler to introduce, for example devices like difficult Chow moving lemma, needed for the development of Chow motives is not necessary,
Compare derived noncommutative algebraic geometry with the terminology in Maxim Kontsevich’s Miami abstract which deals with noncommutative motives which are represented by derived categories representing (derived categories of coherent sheaves on) varieties. No Grothendieck topologies or infinity topoi in the formalism.
http://www-math.mit.edu/~auroux/frg/miami10-abstracts.html
Maxim Kontsevich: Mixed non-commutative motives and open Gromov-Witten invariants
In derived non-commutative algebraic geometry one can define a triangulated category of mixed nc-motives similarly to Voevodsky’s theory, but in much more simpler and direct way. The conjecture on the degeneration of Hodge to de Rham spectral sequence extends to the mixed setting. Also, one define a refined Chern class of an object of a saturated dg-category in an analog of Deligne cohomology. I’ll present also examples finite diagrams in A-model setting, where one can identify periods of mixed motives with some kind of open GW invariants.
From what little I have read of their papers they view the derived category as basically a convenient package of classically interesting data about a variety (and not as a generalization or extension of the notion of a variety).
The classical data are precisely here the cohomological information. Of course, passing from true, abelian setup to the derived setup one looses some information (the abelian category of coherent sheaves reconstructs the variety without loss of inforation). The derived scheme is now represented by the generator in the derived category which will be a dga (so it fits into the Toen-Vezzosi view or into the view of Lurie), and that dga represents a derived scheme which is simpler than the original scheme. Of course in Toen’s framework the original scheme is also an example of derived scheme in his setup, bit of course, richer than the one produced from derived category. The lost information is in fact in good cases only finite. For example, if the canonical sheaf on a smooth projective variety is ample, Bondal and Orlov reconstruct the variety back. Thus most sharp advances in algebraic geometry in last two decades were indeed obtained working with the derived image in this weaker sense. This is a flavour of derived algebraic geometry which perfectly fits into the general second kind but the way one produces and analyses examples is quite different: one stays in the world of “categories of coherent sheaves” rather than looking at an ambient topos into which one can assemble all such. There is also the freedom of noncommutativity which fits more into such applications like homological mirror symmetry and noncommutative motives.
To clarify the additional point, the category of usual schemes embeds into the higher category of derived schemes in the sense of Toen. This embedding does not loose the information. However, the applications like mirror symmetry, associate to a scheme a stable infinity category (= derived category of coherent sheaves or alike) with a loss of information. This image is in the community of noncommutative derived geometry also referred to as a derived scheme. And indeed it is a derived scheme in the sense of Toen-Vezzosi-Vaquie, if the underlying dga is commutative. So we have two functors from schemes to derived schemes, one is faithful, another is not. Two flavours center on which embedding is central viewpoint when extending the usual examples from algebraic geometry. The non-faithful had so far more genuine applications in algebraic geometry and representation theory and is older. Beilinson started looking at derived categories of coherent sheaves in seminal works in late 1970-s. This got continued with works of Kapranov, Orlov, Bondal and others in late 1980s which introduced exceptional collections and other geometric structure at the derived level and the definition of enhanced triangulated categories. Kontsevich refined the picture in A-infinity language with lots of applications in early 1990s. One of the principal motivations of Simpson-Toen school was to create the moduli spaces for A-infinity structures and alike, so to say representing objects for functors in this world. There is also a role of some conjectural pictures from Deligne and from Drinfeld…
The derived algebraic geometry of Toen is mainly based on spaces locally represented by COMMUTATIVE dgas.
And this is the point that the entry doesn’t say clearly, but which it should. It’s not about whether one talks about derived categories of coherent sheaves. It’s the step from formal duals of commutaive rings to formal duals of simplicial commutative rings / commutative dg-rings in negative degree that makes derived algebraic geometry.
The noncommutative aspects should go in some other entry, maybe derived noncommutative algebraic geometry. What the entry derived algebraic currently highlights is not what is commonly understood as “derived algebraic geometry”.
It’s not about whether one talks about derived categories of coherent sheaves
Urs, maybe we should say two motivations. One is from moduli spaces, one is from derived categories of coherent sheaves (and Fukaya etc. categories). But the framework of course may cover lots of examples which are neither of the two. But those are the content of ideas driving the subject and should be central, the concrete formalisms are secondary.
The noncommutative aspects should go in some other entry, maybe derived noncommutative algebraic geometry.
This is impossible to divide. The deformation theory of commutative examples, like comutative varieties involves the study of noncommutative examples. Moduli spaces of noncommutative examples, are in some important cases, commutative derived schemes in the sense of Toen-Vezzosi. It is one subject, there are two principal motivations, and two principal formalisms, but the overlap is big, and the same people worked on both (including Kontsevich, Toen, Kaledin, Katzarkov).
With the same argument it would be impossible to divide noncommutative geometry from geometry.
But even if it stays all in one entry, that entry should state the definition correctly: by “derived algebraic geometry” people mean higher geometry over a site of simplicial/dg/$E_\infty$ rings (op). Nothing else.
But, also A-infinity rings, and they do not form a site (edit: they may, but the extensions of many interesting Grothendieck topologies for E-infinity rings rather extend into things which are not Grothendieck topologies; the noncommutative topology formalisms may apply).
The fact is that at the derived level commutative and noncommutative get quite mixed. The semiorthogonal parts in analysis of commutative schemes give noncommutative exmaples, similarly with moduli spaces. Besides things like noncommutative projective spaces have the same derived category like commutative projective space. As long as you are interested only in some moduli spaces in algebraic topology you do not get to consideration like this because you just want to constuct the moduli space. But if you study algebraic geometry, for example subvarieties in this (moduli) space you will get to study noncommutative as well. It is far more convoluted. And the subject of what you would call commutative is developed largely motivated by examples from noncommutative world. Physicists would not look in 1990s into those if there were not things like mirror symmetry.
On the other hand, I agree that noncommutatyive geometry is also a part of geometry, and it should be reflected in the entry geometry. As Kaledin puts in his Tokyo lectures, noncommutative geometry is a style of doing geometry where the space is replaces by category of objects living on the space and which stems from things like Serre’s 1950s theorems in the case of algebraic geometry.
The categories of (quasi)coherent sheaves derived or not are quite close abelian or stable analogues of topoi or higher topoi. In 1-categorical situation Durov unified them under the notion of vectoid. The reconstruction theorems for schemes are analogues of Giraud’s theorem for topoi. Once one is taking the topos theoretic viewpoint, one is implicitly thinking in the philosophy of noncommutative geometry.
The split is morally much about the choice small site/big site. Small topos of sheaves on a scheme, or rather vectoid of coherent sheaves on a scheme, versus topos of sheaves of sets on a large site. And higher/stack analogues. If we split of the noncommutative point of view we not only loose main applications of derived algebraic geometry in commutative algebraic geometry, but also give a strong preference to gros topos point of view.
Link to Kaledin’s Tokyo lectures: pdf
One can also go up the ladder: for a commutative dga one can look at monoidal stable category of dg-modules and it can be interpreted as a $(\infty,2)$-algebra. Then the modules over it have a monoidal structure again so it is a $(\infty,3)$-algebra and so on. The derived algebraic geometry for such is not yet developed…and it is far beyond the geometry based on simplicial algebras…
I have rewritten the Idea-section to read as follows:
What has been called derived algebraic geometry by (Kapranov), (Toën-Vezzosi), (Lurie) is higher geometry locally modeled on formal duals of commutative simplicial rings, dg-algebras and generally E-∞ rings, hence the derived geometry refinement of traditional algebraic geometry.
More generally one may consider a derived version of noncommutative algebraic geometry, over A-∞ algebras (Kontsevich-Soibelman).
Whereas “commutative” derived geometry over E-∞ rings is well described by (∞,1)-topos theory, the noncommutative flavor needs higher tools.
Eventually the entry should be reworked such as to not make the impression that studying derived categories of coherent sheaves is already what is called “derived geometry”. But I won’t do that right now.
I have replaced reference in that idea sentence to Kontsevich-Soibelman paper to the reference to the Katzarkov-Kontsevich-Pantev paper because only the latter indeed develops full-fledged noncommutative derived algebraic geometry while the first is mainly working on the infinitesimal picture only (what has been written in the sentence before the reference is given, in the literature section). Hence, the deformation one is mainly about noncommutative formal schemes, rather than the formalism of derived geometry in A-infinity context.
not make the impression that studying derived categories of coherent sheaves is already what is called “derived geometry”
By itself, study of enhanced derived categories of sheaves, is just a fragment of the full theory (these examples may be viewed as being at the intersection of commutative a la Toen theory and noncommutative a la Koontsevich), but it does belong there: the generator will be a dga, hence it fits even into your definition. But you can also consider the pretriangulated dg-category (or a stable category) as the structure $(\infty,2)$-sheaf from the start (does number 2 makes you uneasy here?). Though these consideration are early they point toward much more general picture which will require eventually allowing $(\infty,n)$-sheaves for all $n$, what is beyond the framework in Lurie’s book.
Also if you go via enhanced derived categories, then the motives you get are the plain old Chow motives – this is a theorem: the category of Chow motives localized at Lefschetz motive embeds into the category of Kontsevich motives. Thus if you start with a commutative scheme, then taking its noncommutative motive or commutative Chow motive is just a matter of taste, and the noncommutative way is technically simpler.
I added the reference of Tabuada on noncommutative motives.
Okay, thanks.
By itself, study of enhanced derived categories of sheaves, is just a fragment of the full theory (these examples may be viewed as being at the intersection of commutative a la Toen theory and noncommutative a la Koontsevich), but it does belong there: the generator will be a dga, hence it fits even into your definition. But you can also consider the pretriangulated dg-category (or a stable category) as the structure (∞,2)-sheaf from the start (does number 2 makes you uneasy here?).
I am fine with what you say here… except that I keep insisting that this is not what most everboy that I know means by “derived” geometry.
I don’t claim that the now established term “derived algebraic geometry” is particularly well chosen, and I understand that it may sound like referring to all kinds of higher geometry. But that’s not how the inventors of this term (notably Toën-Vezzosi and Lurie) use the term: for them “derived” geometry is about “deriving the site”, passing from formal duals of rings to formal duals of $E_\infty$-rings (or their approximation by dgc-algebras etc.)
The $(\infty,2)$-algebraic geometry that you keep referring to is also a kind of “higher” geometry, but this is not what is commonly called “derived algebraic geometry”.
(Maybe you can argue that it should be called this way, but then the entry should at least make clear that it is proposing non-standard terminology. Nothing wrong with that, of course, if there is a good reason. And maybe here there is a good reason. But this will be an uphill battle, since after Lurie’s 14-piece series on “derived algebraic geometry” it’s a bit tough to just come along and declare that the term is now to mean something else.)
The generalization to $(\infty,2)$-algebraic geometry that you are talking about is a generalization of algebraic geometry in a different direction. Maybe with overlap, but in itself different.
I am all in favor of having entries on this $(\infty,2)$-algebraic geoemtry. I think it’s great and I’d like to eventually dig deeper into it. But I feel it needs to be disentangled form what about everyone these days thinks of when saying “derived algebraic geometry”.
You mean “about everyone” because you are in algebraic topology/higher category circle. Much of the practical algebraic geometry school, especially Moscow, which uses noncommutative derived algebraic geometry informally “derived picture”, ”derived scheme”, ’at the derived level” when talking about varieties in this picture. All of them are very well aware of Toen’s school and are eager to interact (look at the Dmitri Kaledin’s beautiful Tokyo notes to see how the subject is intertwined and unified). You may consider this minority has majority of derived-style results in concrete algebraic geometry – I mean about Fano varieties, Calabi-Yau, connections to tropical geometry and so on. The $(\infty,2)$ is just one point of view, As I said most of commutative examples can be considered as just giving a different picture of a particular derived schemes in the sense of Toen. Of course, if you just want formal picture you can disqualify much of these, but if you look at the practical level it is the same subject.
It is similar to the picture of cyclic homology. I is originally devised for noncommutative geometry, but there are more and more works of essential usage in usual geometry. For example recent paper of Connes and Consani shows very deep connections to L-functions and rather central emerging role in arithmetic geometry in general.
I have created entries (related to the above discussion)
I have to agree with Urs and the others. I’m very early in my career, but my thesis is entirely about the derived category of Calabi-Yau’s, so I’ve talked with and read all of the major figures “doing derived categories.” I’m referring to Caldararu, Huybrechts, Orlov, Lieblich (my advisor), and others. Not once have I ever heard any of them refer to what they do as “derived algebraic geometry” and I’d certainly never say that about what I do.
I admit that some recent results about dg-enhancements and so on bring the subject very close to derived algebraic geometry, and even people like Orlov who have gone in that direction may use the term occasionally, but I’d say he and the others are very careful to only use it in those rare situations where the cross-over actually happens. Most people working on derived categories want to keep a clear distinction between these things.
So why don’t you look at the abstract of Kontsevich’s talk. The general theory is accepted to be called noncommutative derived algebraic geometry, but many examples fit precisely into the commutative derived schemes in the sense of Toen, as the generators are then commutative dgas, as you know. Derived algebraic geometry as a general concept includes the noncommutative one, like general geometry includes the noncommutative one. The fact that people rarely say that is that it is very common that if you do the derived categories of varieties that their deformations, categorical resolutions, mirrors etc. will be noncommutative. So people emphasise on “noncommutative”, what does not exclude it.
And if you were more careful than you would be seeing that I am gradually passing more toward to your terminology, but this needs care and redistribution of material and not boasting on who is your advisor. It also needs more time as I am spending more time to your complaints than to writing $n$Lab entries. $n$Lab looks toward future and wants to be inclusive rather than looking at western school of writing dominance.
And for the end a question to you Hari. You know that the usual schemes embed into all schemes. This commutes with colimits but not with limits, so the intersection theory changes; hence the derived algebraic geometry and the usual one give different intersection theories. My claim is that the derived categories approach does not give the usual one but one agreeing with the derived algebraic geometry. So, now consider the variety, two subvarieties and more or less the setup of Serre multiplicity formula but with relaxing the generic point assumption. The usual algebraic geometry gives here the conditions similar and more general than the situation of Bezout’s theorem where you for sure recall the transversality assumption. Now the derived geometry, has cleaner theory and due corrected fibre products of derived schemes you will get a result without exceptions. Now the Serre’s formula may be said also cleanly in derived categories language. Consider the enhanced categories and the ones for the subvarieties and go toward the corresponding definitions of intersection numbers in those terms. I think you are inevitably getting toward the derived version of the theorem. Id est, once you pass to stable infinity categories of coherent sheaves, the intersection theory must be formulated to give the same results as the theory of commutative derived schemes. Hence you are from the start out of usual algebraic geometry and you are factually in derived. We talk essence, not the formalism here. Do you agree, or you think that there is a (to me unknown) consistent sound way to get the usual algebraic geometry picture of intersections ? (I mean in general case)
The general theory is accepted to be called noncommutative derived algebraic geometry,
Kontsevich and Rosenberg speak of “derived noncommutative algebraic geometry”.
And for good reason I suppose: this is not the Toën-Vezzosi-Lurie-type derived algebraic geometry made noncommutative. It is instead the approach of formulating noncommutative spaces by their categories of coherent sheaves made derived by passing to higher categories of coherent sheaves.
not boasting on who is your advisor.
I don’t think anyone was boasting. Instead, somebody provided evidence for the point under discussion.
nLab looks toward future and wants to be inclusive rather than looking at western school of writing dominance.
This is not about dominance but about not confusing readers with siltently changing terminology conventions. If you insist that your terminology is that of the future you should make the entry say that clearly with something like “Warning: what we call ’derived algebraic geometry’ in the following is not what is currently usually understood under this term, but is rather what Kontsevich and others call ’derived noncommutative algebraic geoemtry’ “.
But what I don’t see is the motivation for conflating terms in this way. I think the term “derived algebraic geoemtry” is unfortunate enough that if I were promoting Kontsevich-style geometry I would want to make sure that people do not confuse it with Toën-Vezzosi-Lurie-style geometry.
Especially if you are afraid of being dominated by approach X, then you shouldn’t declare that your beloved appoach Y is from now on also to be called “X”. Because people will rightly assume that you are talking about X when instead you are talking about Y and as a result you will feel all the more dominated by the inventors of X.
Especially if you are afraid of being dominated by approach X, then you shouldn’t declare that your beloved approach Y is from now on also to be called “X”.
First of all it is not my favourite, I just do not want to simplify the subject (look at the survey by Barwick which acknowledges that the subject originates from ideas of Beilinson, Drinfeld, Deligne, Kontsevich from 1980s, and you can follow what was indeed the state of the subject in 1980s and you will come indeed close to the stuff I am talking about). I did not. I said that there are two subjects but two sources of the subject/examples and that the subject is unique and nonseparable. And I said that I will rework the entry but simply loose energy on your criticism which does not account for the main theorems I quoted (like embedding of commutative motives into derived noncommutative a la Kontsevich).
Kontsevich-style geometry I would want to make sure that people do not confuse it with Toën-Vezzosi-Lurie-style geometry
It is not to be confused: I think there are two correspondences from usual varieties to derived schemes in the sense of Toën-Vezzosi, one is embedding described in Toën’s beautiful introduction lectures on the subject, and the other one is the one which assigns the cdga which is the generator of the derived category. Now you can proceed to spend time talking about it as a stack of some sort,on some site or you can simply analyze your particular family of schemes staying within the enhanced derived category. I mean not spending too much time on the whole big site but working on the concrete representation of a derived scheme. I asked Toën in 2005 about the difference between the faithful and “derived” image and he told me that they have a remarkable general theorem that the fiber between the derived and nonderived embedding is finite (I do not recall if he said what assmptions on the original scheme he assumed). This theorem is a general result which puts Bondal-Orlov result (reconstruction of smooth projective variety from derived category when the canonical or anticanonical bundle is ample) into the general setting (in that case fiber is trivial, hence finite).
I suspect that Hari is working on usual derived categories and not enhanced derived categories (but I don’t know, this is the way mentioned Huybrechts and other people were mainly working in published works known to me). Nonenhanced derived categories do not qualify for derived algebraic geometry (though individually things may be from recently saved a bit using Orlov-Lunts uniqueness of enhacement theorem, which is however using rather difficult machinery). With the usual derived categories you loose some information, which is necessary to do sensible geometry of morphisms of schemes. For example, the Toën’s beautiful survey on the differential graded categories puts as a basic motivation exactly this point: if you replace schemes by the derived categories and you look at the gluing of schemes along localizations, then you miss the descent theorem (because of bad behaviour of homotopy limits). If you, however, do this with the enhanced version of derived categories, e.g. withe differential graded categories things do work. Here is pdf.
If you did not complain so much I would long since make this entry better that you would be long since happy. This way I need to spend energy, and repeat in a different way as I see that you did not get the point: The generator of the derived category is a cdga which inputs into Toën-Vezzosi-Vaquie formalism if you like. People just do not need to go into full formalism to get great applied results for small families of quasiprojective varieties, still at the (enhanced) derived category level. Only things like its resolutions, deformations (needed for mirror symmetry and so on) naturally go beyond commutative dga. Of course, for general results on the collections of all derived schemes you need full formalism, commutative or noncommutative.
Kontsevich and Rosenberg speak of “derived noncommutative algebraic geometry”.
I agree that this order is better (from the point of view of a noncommutative geometer) than indeed more rarely used noncommutative derived algebraic geometry. I was not careful here (as I am used to hear just derived in jargon) and was putting one as a title and another as a redirect, you are free to change (or I will once I go to update it).
Edit: the entry is in fact called (as it was months ago) derived noncommutative geometry with redirects both noncommutative derived algebraic geometry and derived noncommutative algebraic geometry).
with siltently changing terminology conventions
Urs, this was a stub in which I wanted to include first all the important material (I have spent many hours on references etc. for this and related entries like motive, Nori motive, derived category of coherent sheaves, noncommutative motive, hidden smoothness principle, derived noncommutative geometry, dg-category, enhanced triangulated category etc. to be now just ciriticised for days of work), and then to make it more coherent. I was all the time emphasising that there are two sources of motivation and general kinds of approaches (which we now a posteriori agree to call noncommutative and commutative), so nothing was there silent.
This article makes a very nice mixture between the two approaches (following Toën, Ben-Zvi - Francis - Nadler and Lunts). In fact the infinity-geometric function theory in the BenZvi-Francis-Nadler flavour is more or less the synonym for the statements like that the tensor products of (enhanced derived) categories of (quasi)coherent sheaves behaves precisely as the derived fiber products of derived algebraic schemes of Toën-Vezzosi.
I wonder if there is a general nonsense viewpoint on quasicoherent modules which makes the connection between the fiber products of big site higher stacks and the tensor products of higher categories of quasicoherent modules even more transparent. What do you think ?
As far as the derived moduli spaces realized in Kapranov-CiocanFontained and then in Toen-Vezzosi-Vaquie the prediction (without still full formalism) with lots of details on the underlying homological algebra is very clearly in article (especially 1.4.2)
which is motivated by Gromov-Witten invariants, virtual fundamental class and mirror symmetry (who said it does not belong to derived geometry). Over there MK constructs a higher structure sheaf by a formula superimposable on Serre intersection formula. I will put the reference also under hidden smoothness principle. I should also think how to incorporate (into the related entries) the perfect obstruction theory of Fantechi-Behrend, which makes the clean general theory behind the things like virtual fundamental class and alike.
I moved part of the material from derived algebraic geometry into derived noncommutative algebraic geometry. I added some references and improved some text and (cross)referencing at both and some related entries. Not finished yet.
New stubs Barbara Fantechi and perfect obstruction theory.
Dear Zoran, thanks a lot for your extensive comments, which I found very insightful.
I just edited the page to fix a small misattribution: the lecture notes of Vezzosi are actually from Tabuada.
Oh, great, probably this was why I could not find them on the net (I downloaded them and had hard time to reobtain the link :)). Tabuada is a great new name in the field, and made major advances and systematization to noncommutative motives.
I did a lot of reorganizing of
trying to clean up the noncommutative stuff mixed in throughout the page (which can already be found at the page derived noncommutative geometry). I hope the page is a lot clearer now to anyone who wants to know what derived algebraic geometry is.
Also some minor changes to derived moduli space, and new stubs
I updated the Barwick reference at hidden smoothness principle. It seems the IAS has removed a lot of personal webpages recently.
associate to a scheme a stable infinity category (= derived category of coherent sheaves or alike) with a loss of information. This image is in the community of noncommutative derived geometry also referred to as a derived scheme. And indeed it is a derived scheme in the sense of Toen-Vezzosi-Vaquie, if the underlying dga is commutative. So we have two functors from schemes to derived schemes, one is faithful, another is not.
Zoran, I know you repeated this point already several times above, but would you mind elaborating on the construction of this functor a bit? The generator of the derived category is a complex of coherent sheaves, how does one get the “underlying dga”? I think this is the point which should be added to the page, under a heading of “Relationship with derived noncommutative algebraic geometry” or similar.
Also, do you know if this construction is written down or studied anywhere?
For proper smooth varieties one can recover the derived category from $Ext^*(E,E)$, where $E$ is the strong generator (in general we get $A_\infty$-algebra); this is guaranteed by Bondal-van den Bergh; you know this work better than I do. The fact that forgetting is just forgetting finite information I was told by Toen in Cortona several years ago, I never thought in depth about it. As far as functoriality, of course one has nonuniqueness of generators, just like in abelian context; in abelian context however as long as one has a commutative representative one has a unique such choice. I do not know how similar logics of choice works in triangulated context, you should ask the experts for the derived picture.
The difference between commutative and noncommutative approach to derived geometry is roughly like between petit and gros topoi. Toen-Vezzosi and Lurie work with infinity-sheaves of sets on higher site of dg-affine schemes, this is working in gros topos. On the other hand replacement of a small topos is working within (quasi)coherent sheaves on a single variety, in abelian or triangulated sense, so it is not a topos but morally it is. In Abelian context, Rosenberg was working out some theorems of that kind, but I think they were just partially written down (like getting noncommutative schemes via adjoint functors like in his paper on nc schemes and alternatively as sheaves of sets on Q-category which he consider a replacement of Zariski site in nc context).
Thanks! I wrote something here, please check if I understood properly.
Roughly, I believe it is, but I am really not that competent for the derived picture. Thanks.
Maybe we can say this more conceptually. First, some derived schemes in the sense of DAG5 are faithfully encoded by their stable infinity categories of quasicoherent sheaves. That’s the content of DAG8. Now one can turn thi around and declare that a stable infinity category which does not arise from a dag5-scheme is a “noncommutative” such beast. Under some niceness assumption this stable infinity category is presented by an A-infinity category in the linear sense, and this way one recovers Kontsevich-style higher NCG.
Interesting (that this is in DAG8).
As far as the remark about linear or not, in his talks (I heard some 9 years ago, but also in later ones), Kontsevich was emphasising that the linear enrichement for his nc spaces is just a practical approximation, but that he really means enrichment over spectra, rather than special linear case of dgas. On the other hand, he had also a side statement that he expects to have some sort of equivalent operator algebraic picture with nuclear spaces playing their role, if he restricts to nice but large subcategory (corresponding to complex quasiprojective varieties I think).
I see. Certainly the vast majority of publications in this school seem to look at the fragment that can be presented by dg-methods. But of course from the point of view as in #40 it is clear that one can/should consider something more general.
I have now added some paragraph as in #40 to the entry here.
Thanks for the correction.
I have added the today’s reference
As it surveys various approaches, including French school and Lurie, I did not classify it into Toen’s approach but put on top. It is also compact, updated while still technically precise, so it is nice to put it as a general intro reference.
Wow, thanks a lot for that pointer.
I finally corrected the relation between nc and derived ag as mentioned in #43. The correct relationship is an adjunction DSt -> NCSp between the functor associating to a derived stack its dg-category of perfect complexes, and the functor associating to a dg-category the derived moduli stack parametrizing its objects.
Also it seems that DAG over E-infinity rings is actually called “spectral algebraic geometry” by Lurie. Apparently DAG always means over simplicial algebras (or nonpositively graded cdga’s). I have tried to make some edits in light of this.
Yes, he calls it “spectral” but that has n unfortunate clash of meaning. Therefore there is a diamb. Entr spectral geometry
I think it is safe to call this spectral algebraic geometry, just as noncommutative algebraic geometry is not the same thing as what is usually called noncommutative geometry. I see that you are calling this E-infinity geometry; this is reasonable but if you apply the same convention to AG over simplicial/DG-algebras, these should be called then simplicial geometry and DG-geometry. However simplicial schemes and dg-schemes already have different meanings.
I think Urs meant to say “disambiguating entry” spectral geometry in #48. But if not, I’m happy to be corrected.
I agree, there is never a globally good convention. Let’s just try to have relevant disambiguation pointers when we write nLab entries. In a single article it is easy to adopt any one convention, but here where we are discussing lots of different specialities, it’s better to be more careful.
For instance I think that Toen-Vezzosi in their HAG-articles do speak precisely of “dg-geometry”. We have an entry dg-geometry. So E-infinity geometry seems not too bad.
Also, the adjective “algebraic” has its problems. As soon as one starts to merge Connes-style ncg with non-commutative “algebraic” geometry the terms “spectral algebraic” become ambiguous once more.
it is safe to call this spectral algebraic geometry,
No, it is not ! Not only spctra of Laplacians and Dirac operators but also spectra of operator algebras extend beyond operator algebraic context. There are spectra and their elaborate geometry (structure sheaves, structure stacks, geometric morphisms, cyclic homology, coherent sheaves etc.) not only for operator algebras but also for various classes associative (possibly noncommutative) algebras, abstract noncommutative schemes, abelian categories, exact categories and so on. They nontrivially extend the notion of prime spectrum of a ring, cf. entry spectral theory. This is an extensive subject which has preceded noncommutative geometry a la Connes (say works of P. Gabriel, P.M. Cohn, Golan, Van Oystaeyen and others in 1960-s and 1970-s). Therefore for lots of people, specially in “my” subject, spectral algebraic geometry means something different. Also, what is very important, the study of spectra of operator algebras and other topological algebras and of non-topological algebras and some other kinds of algebras are all quite related and it would be non-natural to separate those. by allowing spectral geometry terminology for one and not for another.
It is worthy to mention also that the algebraic geometry over spectra of algebraic topology has been called new brave algebraic geometry by Peter May and was quite an accepted term already a decade ago. Of course, introducing clashing terminology is allowed to mathematicians who made important work (and in contemporary hyperproduction of new elaborate abstract formalisms almost impossible to avoid), but calling this terminology “safe” is not justified.
By the way, we also have an entry brave new algebra. In case anyone feels that’s worth pointing to for clarification in some entry.
I added the sentence
Most of those belong either to the geometry as seen either by point spectra of spectral theory (of operators, families of operators, operator algebras, rings, associative algebras, abelian categories etc.), or by spectra in the sense of stable homotopy theory like symmetric spectra, $E_\infty$-spectra, ring spectra…
to spectral geometry and updated discussion in 52 above.
For instance I think that Toen-Vezzosi in their HAG-articles do speak precisely of “dg-geometry”. We have an entry dg-geometry. So E-infinity geometry seems not too bad.
Our entry on dg-geometry is what Toën-Vezzosi call complicial algebraic geometry (from “complex”). In the literature dg-scheme refers to something different, which is just a historical precursor to the correct notion of derived scheme. Kontsevich and Toën have both suggested forgetting this notion entirely, but there are still important papers of Ciocan-Fontanine and Kapranov where they construct eg derived Hilbert scheme and an explicit translation into DAG still hasn’t appeared as far as I know. So in my opinion it is still too early to start using the term dg-geometry without risk of confusion.
It is worthy to mention also that the algebraic geometry over spectra of algebraic topology has been called new brave algebraic geometry by Peter May and was quite an accepted term already a decade ago.
This should be the same as the brave new algebraic geometry considered in HAG II. This is the third HAG context they consider, but it is not the same thing as spectral algebraic geometry. If I recall correctly spectral AG doesn’t fit into the TV framework (it’s not a HAG context).
So I think disambiguation pages are a good idea, but still I would prefer to stick to the standard usage of terms (complicial and spectral vs DG and E-infinity) because disambiguation pages are necessary in either case, and there is less risk of confusion anyway between spectral algebraic geometry and spectral theory, than between dg-geometry and the older notion of dg-scheme.
re#55 Woops, you are right, I managed to trick myself here, first changing the terminology and then forgetting that it was me who changed it. Sorry.
So I forgot about this discussion and accidentally duplicated the page E-infinity geometry at spectral algebraic geometry. Can we delete the latter page somehow?
Coming back to the discussion about good termionology for geometry modeled on formal duals of $E_\infty$-rings (from earlier this year above):
wouldn’t higher arithmetic geometry be a good term? (or maybe “arithmetic higher geometry”)?
I mean on absolute grounds, independent of whether it’s wise to use a different term than already established.
What is the connection with arithmetic geometry?
I mean, arithmetic geometry is about the formal duals of $\mathbb{Z}$-algebras aka commutative rings, whereas here we are dealing with formal duals of $\mathbb{S}$-algebras, aka $E_\infty$-rings.
arithmetic geometry is about the formal duals of $\mathbb{Z}$-algebras aka commutative rings
But isn’t this a description of algebraic geometry in general?
So also the term “arithmetic geometry” refers to just algebraic geometry, but highlighting the fact that one will not restrict attention to algebraically closed fields, not to characteristic 0 and possibly not even to fields. It’s still “algebraic geometry”, but with the emphasis that the “most fundamental” variant is meant.
Similarly here: derived algebraic geometry over $E_\infty$-rings is just part of derived algebraic geometry, but one needs a word to highlight that one will consider $E_\infty$-algebras not in characteristic 0 – where they give just the dg-algebra sector of DAG – but that one really means the “most fundamental” version of the theory over general $E_\infty$-rings with no simplifying assumptions.
Ok, I see what you mean. I’ve always understood arithmetic geometry to refer specifically to characteristic p methods. And I would probably use “complex algebraic geometry” for characteristic zero. That’s why I prefer either the term spectral algebraic geometry or E-infinity geometry.
Also I am not sure that this is the most general or fundamental version of algebraic geometry one could consider (which is what the name “higher arithmetic geometry” might suggest). Perhaps one could take different categories than spectra, for example motivic spectra (which would allow viewing the algebraic K-theory spectrum as an affine scheme, and studying its K-theory or other invariants…).
Okay, the “fundamental” argument is too vague to be useful, admitted. But “spectral geometry” is an unfortunaley ambiguous term for such a profound concept. And about mysuggestion “$E_\infty$-geometry” it was actually you who raised complaints above!
Moreover, if “arithmetic” makes you think “positive characteristic” then that’s pretty good already, I’d say, since it makes clear already that we are not talking about dg-algebras. And “arithmetic” doesn’t suggest anything motivic, does it?
So it seems you should actually be happy with the suggestion of “arithmetic higher geometry”!:-) But of course it was just a thought. If it doesn’t resonate with you or anyone, Iwon’t insist.
I actually find the term “E-infinity geometry” good, “objectively” speaking, my only complaint was about the disagreement with the literature.
Oh, I see. Okay.
64 “Complex algebraic geometry” is not a good way to describe characteristic 0. Most characteristic 0 stuff cannot use anything from complex geometry. I think what Urs was getting at is that doing geometry over $\mathbb{Z}_p$ or even $\mathbb{Q}$ is clearly arithmetic, but it lives in characteristic 0.
Yes, but maybe I didn’t say it well. Shouldn’t one really think of it like this: arithmetic geometry is fundamentally geometry over $\mathbb{Z}$, hence with no condition on the commutative rings whatsoever. Then looking at $\mathbb{Z}_p$ and $\mathbb{Q}$ is “just” a way to decompose this difficult subject into slightly less difficult subjects.
In this vein, since the sphere spectrum $\mathbb{S}$ is the homotopy-theoretic refinement of $\mathbb{Z}$, we are led to saying that “higher arithmetic geometry” (or “derived”, if you insist) is geometry over $\mathbb{S}$, hence with $E_\infty$-rings without any further conditions. Again, you may (and you will) go and decompose this into problems over $\mathbb{Z}_p$ and $\mathbb{Q}$, but as before, this is more a “tool” than an end in itself, the subject as such is really geometry over $\mathbb{S}$. Hence “homotopy arithmetic”.
I added links to video recordings of Lurie’s 2014 UOregon Moursund Lectures at spectral algebraic geometry.
Thanks!
I added the notes
to some of the relevant pages.
added publication data for
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