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While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative geometry that behave very much like commutative geometry in this respect. The archetypical example here is supergeometry.

One may argue that the reason that the theory of supergeometry proceeds in close analogy with ordinary differential geometry is simply because a supercommutative algebra is just a commutative algebra, but internal to a nontrivially braided symmetric monoidal category. On the other hand when it comes to local properties and the fact that Grothendieck topologies work well in supergeometry, this is to do more specifically with the fact that the non-commutativity is all in the nilpotent ideals of supercommutative superalgebras, and hence irrelevant to coverings and descent.

This leads one to wonder whether there is something to be gained in developing a geometry based on formal duals to those noncommutative algebras for which "all the noncommutativity is in the nilpotent ideals", e.g. such that when quotienting out the maximal two-sided nilpotent ideal they become commutative. Supergeometry would be a special case of this, but it would be more general.

Has anything like this been investigated somewhat systematically anywhere? Is there any names attached to this that one could search for to find more?

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  • $\begingroup$ what are other examples of such NCGs? $\endgroup$ – bananastack Apr 1 '15 at 5:23
  • $\begingroup$ Cross posted from MSE math.stackexchange.com/questions/1214990/… $\endgroup$ – David Roberts Apr 1 '15 at 7:27
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    $\begingroup$ @bananastack: a natural class of examples is provided by truncations of the homotopy groups of a connective $E_n$ ring spectrum, $n \ge 2$ (e.g. the "group algebra" $\mathbb{S}[\Omega^n X]$ of the $n$-fold loop space of a space). Studying the noncommutative geometry of these things should be an approximation to studying "$E_n$ algebraic geometry" itself. $\endgroup$ – Qiaochu Yuan Apr 1 '15 at 8:04
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    $\begingroup$ In line with Qiaochu's comment, some aspects of E_n-geometry are studied in the thesis of John Francis and this sequel. $\endgroup$ – AAK Apr 1 '15 at 9:18
  • $\begingroup$ Thanks, excellent. I suppose I should have been aware of this. The thesis in chapter 4 has just what I have been after here. Did anyone have any substantial thoughts on the perspective of regarding this as generalized supergeometry? For instance Deligne's theorem ncatlab.org/nlab/show/Deligne+theorem+on+tensor+categories is suggesting higher versions with symmetric monoidal 1-categories replaced by symmetric monoidal (oo,n)-categories. Deligne finds that supergroups are the Tannakian dual for n=1 (subject to a nicety condition). Can one find "E_n-groups" as the dual in the higher case? $\endgroup$ – Urs Schreiber Apr 1 '15 at 10:13
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Lieven le Bruyn kindly points out

  • M. Kapranov, Noncommutative geometry based on commutator expansions, J. reine und angew. Math. 505 (1998), 73-118, math.AG/9802041

which develops pretty much exactly the idea that I was asking about. With that in hand, Google tells me to my surprise that my own wiki had a hidden entry on this all along

which Zoran Škoda once started, thankfully. This has a few more links. Good stuff.

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