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Added to noncommutative algebraic geometry a section “Relation to ordinary algberaic geometry” with what is really just a pointer to an article by Reyes:
The direct “naive” generalization of Grothendieck-style algebraic geometry via sheaves on a site (Zariski site, etale site etc.) of commutative rings-op to non-commutative rings does not work, for reasons discussed in some detail in (Reyes 12). This is the reason why non-commutative algebraic geometry is phrased in other terms, mostly in terms of monoidal categories “of (quasicoherent) abelian sheaves” (“2-rings”).
This is not entirely true: if Reyes considered analogues of Zariski, etale etc. But there is now e.g. smooth topology on NAff which is a genuine Grothendieck topology. How good is this or not it is dictated by supply of relevant examples, not by abstract theorems. Also, more widely, one can consider sheaves on Q-categories, then the Zariski works as well. Finally, Reyes does not take into account Morita bimodules in any way, that is maybe why his matrix rings give problems.
BTW, your link to Reyes in 1 does not work at the moment.
The link to Reyes does work from within the entry. The blue box above is entry code copied to here.
The blue box above is entry code copied to here.
I see, then it should be modified, so that the appearance is the same but the link works. Now it looks like a link, but when you click you get nothing.
Sorry, there is a limit to the extra work I will do beyond announcing an entry. I work on the entry, add hyperlinks, cross-links, TOCs, then copy all my additions to here for you to see. I don’t have the energy to create here a full working duplicate of the entry on top of it all. You are please asked to regard my copying here of my edits as extra service and otherwise please check out the entry itself.
Surely, Urs, nobody expects you to do “full working duplicate”; this whole $n$Forum discussion thread is about results of one single paper, so I called only for one single key link needed to make the thread functional (especially so often when $n$Lab is down or on slow connection):
You please also regard my frequent and altruistic feedback to the edits and posts of your momentary interests extra service to your work and interest – reply 5 discourages me to spend my time for such feedback.
I added the very nice new preprint
to noncommutative scheme and derived noncommutative algebraic geometry.
Wow, quite a treatise! And well written.
Following discussion here I noticed that the section “Derived non-commutative geometry” here barely mentioned the key points.
Don’t have time to do this justice now, but I have expanded just a little to make sure that at least some key facts are mentioned. Here is what it says now, please feel invited to expand (also somebody should clean up the whole entry, but I don’t expect anyone will do it…):
Any locally presentable stable (∞,1)-category may be regarded as a stable (∞,1)-topos (the “stable Giraud theorem”, see there). Therefore, thinking of a non-commutative space as the formal dual to a triangulated category or rather to a stable (∞,1)-category is directly analogous to the way topos theory and in particular higher topos theory characterizes generalized spaces as formal duals of toposes.
More in detail, every stable (∞,1)-category with a set of generators is equivalently the $\infty$-category of (∞,1)-modules over an A-∞ algebra (or rather $A_\infty$-algebroid, in general, due to Schwede-Shipley 01) and in this sense manifestly the formal dual to a non-commutative derived variety.
The notion of noncommutative motive as localizations of stable $\infty$-categories (see there) due to Blumberg+Gepner+Tabuada 10 directly ties into this.
Oh dear, now I see that we also have derived noncommutative geometry which also doesn’t say this clearly. But I leave it at that now, have other things to do.
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