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 if 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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