Can someone tell me what's the appalication of the Rosenberg' method on the definition of
the noncommutative algebraic geometry (left spectrum)?
Now when talking about noncommutative algebraic geometry, we will follow the way of the
category theory (abelian category, triangulated category and so on). What's the obstruction of the direct way from the algebraic geometry rather than the category theory?
For example, have someone studied or founded the theorem like RiemannRoch theorem on the noncommutative schemes?
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Quick answer (to the second part): noncommutative rings don't have enough ideals to make a decent spectrum. In words of Fred van Oystaeyen: "it doesn't matter how you try to define what is a point of a noncommutative space, you never have enough of them". There were some attempts of following more classical lines in the late seventies and earlymid eighties (see papers by Van Oystaeyen, Verschoren, and many others) but eventually everybody agreed that a more abstract approach was needed to get meaningful geometric information. 

