TITLE: Non-commutative algebraic geometry
QUESTION [57 upvotes]: Suppose I tried to take Hartshorne chapter II and re-do all of it with non-commutative rings rather than commutative rings. Is this possible? Which parts work in the non-commutative setting and which parts don't? 
Edit: I also welcome any comments/references regarding any reasonable notions whatsoever of "non-commutative algebraic geometry".

REPLY [6 votes]: Rosenberg and Kontsevich will publish three volumes books soon. I have got the second draft of these books which Rosenberg has already taught us. There is a historical observations in the preface of the first volume of these books. I posted here:
Historical observations on noncommutative algebraic geometry I
Historical observations on noncommutative algebraic geometry II 
Historical observations on noncommutative algebraic geometry III

Historical observations on noncommutative algebraic geometry IV
Historical observations on noncommutative algebraic geometry V
Pseudo Geometry I II III

REPLY [3 votes]: As mentioned above, the lack of localization will be a major issue. It is interesting to note that many homological constructions will work. For example, major work was done by Van den bergh, Zhang, Yekutieli and others in constrtuing dualizing complexes over some noncommutative  rings, thus allowing some of the features of the Grothendieck duality to be recovered. See for example https://www.jstor.org/stable/2161576