The algebro-geometrical version of K-theory

Question

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.)

Answer 1

When $R$ is regular, you can use the Karoubi-Villamayor construction, which has a less topological flavor than Quillen's and avoids mentioning CW-complexes.

Consider the simplicial complex that has, in the $n$th place, the group $$GL(R[t_0,\ldots,t_n]/\Sigma t_i=1)$$ and with face maps given by setting various $t_i=0$. The homotopy groups of this complex are isomorphic to the Quillen $K$-groups. Unfortunately, this doesn't work so well when $R$ is not regular.

Answer 2

Instead of CW complexes you can also use simplicial homotopy theory (as explained in Goerss-Jardine's book of that title or Hovey's book on model categories) and define homotopy groups in the combinatorial style as pioneered by Kan. This also gives an approach entirely avoiding the word "topological space". You would just use things dear to any algebraist's heart.

The space $\operatorname{Spec}(R)$ (with the Zariski topology) will not be of much help for what you want. Indeed, for any field this is always just a single point, but the K-theory of different fields varies drastically. Thus, at the very least you will have to use the full scheme or full locally ringed space structure.

Also switching from the Zariski to the ´etale topology (or Nisnevich...) will not help you much, for various fields with the same absolute Galois group will still have distint K-theory (e.g. finite fields and the Laurent series over the complex numbers).

Generally, the way how the Zariski topology plays a role in algebraic geometry is very different from the role of topology in, say, the Q-construction or plus construction. For example the i-th K-group being the i-th homotopy group has not a very immediate link to the Krull dimension of the ring in question. Again, this can be seen for fields. They have Krull dimension zero, but usually all higher K-groups are nonzero. If you take stuff like the Galois cohomological dimension (that is: "etale dimension" essentially) into account, there is a little bit of a link, but to understand that, you would already have to use a lot of deep maths (I am referring in vague terms to the [nowadays proven] Quillen-Lichtenbaum conjecture)

Answer 3

Another geometric approach to $K$-theory (assuming you are familiar with $K_0$) is given in the paper:

Bloch, Spencer: An elementary presentation for $K$-groups and motivic cohomology, in Motives (Seattle, WA, 1991), pp. 239–244, Proc. Sympos. Pure Math., 55, AMS, Providence, RI, 1994.