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Noncommutative algebraic geometry is a developing field. Things have not yet got the final form as in commutative geometry.

But one might wonder whether things are any better in the case of skew-fields, ie division rings, ie possibly noncommutative rings in which each nonzero element has a multiplicative inverse. Algebraic geometry is much simpler in the case of fields. So are things better in the case of skew fields? For instance, is there any particularly nice geometry over the quaternions, like there is one over the complex numbers?

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I feel guilty adding this as an answer, in the type of NCAG that I do(a la Rosenberg) we work in a categorical setting, where we abstract many of the notions well beyond the concept of abelian-ness. We work in non-abelian categories to generalize the geometric ideas of algebraic geometry using the philosophy of Serre/Grothendieck that we need only QC sheaves on a space, not the space. In this sort of approach, skew fields would be a special case. To get a feel for the approach, consult my answer here mathoverflow.net/questions/10512/… . –  BBischof Aug 9 '10 at 23:02
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Such long comments look ugly. If you feel guilty, why not add your answer as community wiki? That will reduce the burden on your conscience and it will look pretty. If you do that you can delete your comment and I will delete mine. :) –  user977 Aug 10 '10 at 0:19
    
but then people will think someone has answered your question rather than painlessly avoiding it while shamefully self-promoting... –  BBischof Aug 10 '10 at 2:13

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