It has been proved that the higher K groups of a (possible noncommutative, but here only comm. for convenience) ring $R$ are correctly defined by Q-construction or + construction.

Recently I'm learning some algebraic K-theory. But I guess I am more like an algebraic-geometor, hence not feeling good about the definition relying on CW complexes. So I am thinking if we can possibly define the higher K groups via the natural space $\mathrm{Spec}~R$. I was somehow told that they are very different. So can anyone can briefly give some reasons why they do not match (and how to construct them algebro-geometrically), as well as some prediction/ideas if we can modify one of the definitions so that they match up.

(BTW, I do not believe that the non-Hausdorffness is the essential reason. We lost some good properties of the topological spaces, but we also have more algebraic structures.)

spectrum? The use of the word spectrum here is wholly unrelated to the usage in algebraic geometry. $\endgroup$ – Denis Nardin Sep 26 at 6:15