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As I see this is a guide to the DERIVED noncommutative algebraic geometry literature only. This should be reflected in the title.
but it seems to me that all noncommutative algebraic geometry is “derived” these days
Is this is a reality show competition or a serious discussion ?
So from my perspective the adjective derived is redundant. Am I wrong?
Absolutely.
Also, when deriving one looses some information and makes things easier. Read, say, Arfin Laudal (e.g. pdf), Michael Artin, Zhang, Smith, van Oystaeyen (e.g. Algebraic geometry for associative algebras, M. Dekker, New York, 2000), Rosenberg (pdf1, pdf2, pdf3). On situation with actions see my paper Some equivariant constructions in noncommutative algebraic geometry, arXiv. Some of the important issues are sheaves on noncommutative topologies (like Q-topology), localization theory, difficulties with morphisms, differential operators, inadequate pullbacks, noncommutative spectra…
The articles you mention are relatively old.
One should familiarize with the field starting with standards, not exotics.
but these are just expositions of work done with Kontsevich decades ago
No.
Isn’t isomorphism in this sense, that is equivalence of abelian categories, too strong to be useful?
There is a field called representation theory. You can be interested in category of representations, or only in its derived category; NAG is largely motivated by it. The derived information is much easier; many things pass easier at the derived level.
Say the application of abelian level NAG are supposed to be the Kazhdan-Lusztig conjectures. They should be an instance of a noncommutative Grothendieck-Riemann-Roch theorem. Crystal bases for quantum groups can be explained from the spectral theory with much new information.
In the triangulated setting, isomorphism seems to be closely connected to geometry, for example as known for abelian varieties, K3 surfaces, etc.
The point of noncommutative geometry is not only that K3 surface has a geometry which can be viewed from new point of view but also that some genuinely noncommutative objects, like noncommutative flag varieties have nontrivial geometry. One wants to view quantum groups as geometrically as one does with Lie groups. Complex commutative algebraic geometry is an important subject but not the only measure of things and not the only source of examples and problems.
Isn’t isomorphism in this sense, that is equivalence of abelian categories
One wants even more: to keep the information on the generator/structure sheaf; keeping the information on the Morita equivalence class.
I will take your preprint
This is Georg. Math. J. 2009. There has been much progress since than in this area, but most is not publicly available. Hopf algebra community is searching for generalizations of torsor theory (related to Hopf algebras, Hopf algebroids, quasiHopf algebras, weak Hopf algebras, coring etc.) and there are hundreds of papers on the subject in last 20 years; this paper just suggests how to go beyond the essentially affine cases.
Did you read any Majid’s work for example ?
The point of noncommutative geometry is not only that K3 surface has a geometry which can be viewed from new point of view but also that some genuinely noncommutative objects, like noncommutative flag varieties have nontrivial geometry.
Ok, but these genuinely noncommutative objects can be studied both as noncommutative spaces or as derived noncommutative spaces. This is not the same thing, so the question is, which is what one really wants to do? “Derived” NCAG isn’t supposed to be a new subject, rather it is supposed to be a new framework for studying the same objects as previous flavours of NCAG. Evidence for it being a better framework is given by the fact that, for commutative spaces like K3 surfaces or abelian varieties, it gives an interesting notion of isomorphism, which is very geometric in nature. In underived NCAG, I thought isomorphism was just the same as isomorphism as commutative spaces. From your second comment, I guess that’s wrong, but still I wonder whether Morita equivalence has some relation to geometry like triangulated equivalence does.
Did you read any Majid’s work for example ?
No, I never even came across it before.
Anyway, I don’t mind changing the title. It’s just that I don’t see “derived” NCAG as a new subject, but rather a new approach to the same one. It’s not analogous to AG vs. derived AG, where one really does study a wider class of objects.
Added stubs for
It’s not analogous to AG vs. derived AG, where one really does study a wider class of objects.
Surely, I was writing this in entries couple of years ago. Derived picture is typically just loosing a bit information, like passing from abelian to derived category. But this is possible just for a PART of NAG, there are many objects in NAG which are NOT represented by abelian categories but other kinds of objects and deriving is not known there.
But it is wider in the sense that not all dg etc. things come from abelian things.
but rather a new approach to the same one.
Approach just to a partial picture of the same one you could say.
I added the reference
to guide to noncommutative algebraic geometry literature and to model structure on dg-categories. On the latter page I noticed the section on the Morita model structure was lacking so I quickly added some remarks.
Thanks a whole lot for expanding that section on the Morita model structure!
(I had added the minimum stub that was there before a while back, when I became aware of Lee Cohn’s article. Didn’t have time to expand further, but clearly this is an important point to expand on.)
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