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!

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    $\begingroup$ Not an answer, just a thought. Schemes were not defined as rings: one started with useful geometric objects, and it took Grothendieck's genius to realize that they are the same as rings. What are the useful geometric objects behind your definition? $\endgroup$ – Alex Degtyarev Mar 5 '14 at 22:36
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    $\begingroup$ Bad for what? What do you want to prove at the end of the day? I think this question is philosophical rather than mathematical. For what it's worth most people seem to think that one should think of some category of modules over the algebra as the space. However it is hard to formulate what is a coherent module for an algebra which is not coherent. So one thinks of the dg-category of complexes of modules instead, where one can formulate the notion of perfect complex. Of course, unlike in the commutative case this is not a tensor category so it is somewhat less geometric. $\endgroup$ – user36931 Mar 6 '14 at 5:06

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.

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 or toposes. 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.

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.

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. 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.

or various other preprints of Alexander Rosenberg.

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

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  • $\begingroup$ This is probably one of the best answers I've seen (in the sense that it answers the question very well), possibly because these results are so good. $\endgroup$ – Harry Wilson Jan 28 '16 at 21:49

You could define a non-commutative affine scheme as the opposite category of a suitable category of rings.

However, if you would like to think about prime ideals as points there are some problems. Namely, non-commutative rings tend to have very few prime ideals, so the geometric structure is to a large extent lost.

On the other hand, a large class of algebras where a reasonable definition would be as you suggest is the class of polynomial identity algebras. These have more or less the same primes as their centres.

Instead a natural way to define nc-schemes is as a category of modules over the ring in question - where the closed points are the simple modules.

You should have a look at (if you haven't already) van Oystaeyen and Verschoren's work from the '70's, and more recently, the school of Michael Artin, including Michel van den Bergh, Paul Smith, John Tate and many many others (although they are primarily interested in projective non-commutative schemes).

At the end of the day, as the comments above suggest, it is what you intend to do with your definition that decides what is an appropriate definition.

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