It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative setting. So my question is:



> 
> Why can't we define the category of "affine non-commutative schemes" by simply putting it to be $\mathsf{Alg}\_k^{op}$, the opposit of the category of unital $k$-algebras?
> 
> 
> 


I was told that it is not a reasonable way of defining non-commutative schemes, but I don't really understand why. Can someone, please, explain to me, why this approach is bad?


I hope this question is not too silly.


Thank you very much for your help!



Why is "naive" definition of non-commutative spectrum bad?
I have also wondered about this question and recently came across some papers that seem to answer it.


First of all, the paper


* Manuel L. Reyes, *Obstructing extensions of the functor Spec to noncommutative rings*, Israel J. Math. 192 (2012), no. 2, 667–698, [arXiv](http://arxiv.org/abs/1101.2239).


shows that there are problems with extending the Spec functor from commutative rings to noncommutative rings.
In particular, any functor Spec : Rings -> Spaces whose restriction to CRings coincides with the usual spectrum functor must map the matrix algebras $\mathrm{M}\_n(\mathbf{C})$ ($n \ge 3$) to the empty space.


One can still hope to have a spectrum in the category of [locales](http://ncatlab.org/nlab/show/locale) or [toposes](http://ncatlab.org/nlab/show/topos).
However, the following paper shows the same result for functors valued in either of these categories.


* Benno van den Berg and Chris Heunen, *Extending obstructions to noncommutative functorial spectra*, 2012, [pdf](http://www.cs.ox.ac.uk/people/chris.heunen/publications/2012/nogo/nogo.pdf).


Finally, one can try to represent noncommutative rings by sheaves on Rings with respect to some subcanonical Grothendieck topology that extends the Zariski topology.
This also fails, as described in the paper


* Manuel L. Reyes, *Sheaves that fail to represent matrix rings*, 2014, [arXiv](http://arxiv.org/abs/1211.4005).


However this approach can be made to work, after modifying the notion of descent.
For this, Rosenberg has developed the notion of a [Q-category](http://ncatlab.org/nlab/show/Q-category).
He realized that, though there is no suitable Grothendieck topology on Ring, it does admit the structure of a Q-category.
See, for example,


* Maxim Kontsevich, Alexander Rosenberg, *Noncommutative spaces*, 2004, [pdf](http://ncatlab.org/nlab/files/KontsevichRosenbergNCSpaces.pdf).


or various other preprints of Alexander Rosenberg.


Interestingly, the work of Rosenberg predates the above "no-go" results by about a decade.


