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I just took a look at the nlab entry: Nikolai Durov. It seems that Skoda never mentioned that what Durov introduced was a special case of generalized scheme theory. I did not read his dissertation carefully or completely. I wonder whether his "generalized scheme" is a special case of noncommutative scheme in the sense of A. Rosenberg.

I will give a talk on noncommutative schemes in a few days. Now I am collecting interesting examples of noncommutative schemes (quasi schemes). Examples that I already know are:

commutative schemes; D-modules; quantum D-modules; almost schemes by Gabber; general Grothendieck category (abelian category); Artin-Zhang noncommutative projective schemes; quantum flag variety in the sense of Rosenberg and Lunts; holonomic D-modules.

Thanks!

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    $\begingroup$ The chief difficulty about reading it carefully is that it has 560 pages or so. $\endgroup$
    – Anweshi
    Commented Jul 27, 2010 at 15:36

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No, there is no need for the Rosenberg noncommutative scheme that the categories be abelian in general; whatever being said in his 1998 paper he does not mean so. He defines a relative scheme over a base category given by exactness properties of direct and inverse image functors describing covers used for gluing and the affinity property. In general one has to be careful, with formulating correctly the exactness properties for nonabelian context.

Of course, to justify this one needs to say that the Durov's schemes are determined by the categories of quasicoherent modules. I do not know if the reconstruction theorem a la Gabriel-Rosenberg holds for the generalizes schemes of Durov. Durov glues schems along categorical localizations which he calls for some reason "pseudolocalizations". They just have the correct exactness properties. On the other hand, for commutative monads, Durov has two nice versions of prime spectra; now one should compare those with a version of Rosenberg's spectrum for nonabelian context. Now one version if the spectrum for right exact categories of Rosenberg. What is a right exact structure for the case of categories like the ones in Durov's work ? Well Durov has spent some time to develop a theory of vectoids which generalize topoi, but also the categories of modules over finitary monads in Set and, more generally, the categories of quasicoherent sheaves of $\mathcal{O}$-modules over generalized schemes. Durov wrote a draft text in Russian about vectoids (which I have seen but is not released yet), and a video of a talk at Steklov. Is there a canonical right exact structure on a vectoid for which the spectrum of the right exact category in the sense of Rosenberg gives a sensible reconstruction theorem, it would be very interesting to investigate.

Remark: I believe that that the quantum flag variety of Lunts-Rosenberg is isomorphic in the case of $SL_n$ to the flag variety studied in my thesis from the dual point of view and with explicit Ore localizations. I never had time to check and publish all the details.

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  • $\begingroup$ very helpful! I am also thinking about whether Rosenberg's spectrum of right exact category gives this reconstruction theorem. $\endgroup$
    – Shizhuo Zhang
    Commented Jul 27, 2010 at 14:36
  • $\begingroup$ Zoran, can we do reconstruction theorem using category of vector bundles on some schemes?, i.e. Vect(X), it is an exact category $\endgroup$
    – Shizhuo Zhang
    Commented Jul 28, 2010 at 5:22
  • $\begingroup$ You can construct some space, but the spectrum will be too small to recover the original scheme; so it is not a reconstruction. $\endgroup$
    – Zoran Skoda
    Commented Jul 29, 2010 at 11:20
  • $\begingroup$ The reconstruction works for Durov's schemes if you take into account the tensor structure (arxiv.org/abs/1202.5147). $\endgroup$
    – Martin Brandenburg
    Commented Oct 5, 2013 at 19:25
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If I recall correctly then Rosenberg's notion of noncommutative scheme is built on abelian categories and then the answer is no: The point of Durov's work is to treat the infinite fiber of Spec Z at an equal footing with the the others and addition breaks down there.

E.g. every monoid yields an affine generalized scheme in Durov's sense, the "modules" over a monoid are sets with an action from it and thus do not form an additive category; homological algebra which seems to be the base of Rosenberg's noncommutative scheme theory, does not exist and has to be replaced by homotopical algebra...

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