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It is known that a quasi compact scheme(even quasi separated scheme)can be determined uniquely by the category of quasi coherent sheaves on it by Gabriel-Rosenberg reconstruction theorem

The elementary case will be the following statement: If we have $R-mod\cong S-mod$(categories equivalence)for commutative ring $R,S$,then $R\cong S$.

However,this is not true for noncommutative ring or more general,noncommutative scheme.The counter example is provided by Morita equivalence.It is know that $R-mod\cong M_n(R)-mod$(as left modules)where $M_n(R)$ is ring of matrices with coefficient in $R$.This shows that category of left modules(considered as category of quasi coherent sheaves on "noncommutative affine scheme")can not uniquely determined the noncommutative ring uniquely.

My question is can we reconstruct noncommutative ring or noncommutative scheme(for nonaffine situation,I am not very sure what I mean,since I dont have an example in my head right now),or more precisely,taking consideration of the examples about $R-mod\cong M_n(R)-mod$,what more information do we need to uniquely determined the ring.

The real motivation for this question is from reconstruction of stacks.I was told by Sasha Rosenberg a couple of years ago that doing reconstruction of scheme,we just need one category,but doing reconstruction of stacks,we need two categories.(Something like fibered category or more general,category over category).It seems that the reconstruction problem for noncommutative ring is similar to that of stacks.So it seems that if we consider certain fibered category,we can do reconstruction for noncommutative scheme.What is the correct formalism to do this.There was a unpublised preprint by Sasha Rosenberg on homological algebra on fibered and cofibered category,he defined spectral of fibered category,but since its highly abstraction,I dont understand much right now

Thanks!

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For stacks a reconstruction theorem was proved in arxiv.org/abs/1004.3087, though this is more in the spirit of Balmer than Rosenberg. In the noncommutative setting, the first question is how to define the categories of quasi-coherent sheaves; this is done in the setting of fibred categories, or categories over a category in Kontsevich-Rosenberg's paper "Noncommutative stacks". I haven't seen any reconstruction theorems in this setting, but this unpublished preprint you mention sounds promising. Wouldn't that be a satisfactory answer to your question? (Is it available online btw?) –  Adeel Jan 3 at 23:57
    
Thank you for pointing out the references.I will take a look at.No,the paper I mention is not available online. –  user41650 Jan 4 at 0:32
    
The questions mathoverflow.net/questions/16257/… and mathoverflow.net/questions/3446/tannakian-formalism deal with related questions - in particular Lurie proved you can recover geometric stacks from their quasicoherent sheaves (not just schemes). In the noncommutative situation the same holds by [a common] definition - a noncommutative affine scheme is a ring up to Morita equivalence (i.e., it IS the category of modules) just as a stack is a presentation up to equivalence. –  David Ben-Zvi Jan 4 at 0:51

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