Geometric Meaning of Different K-theories

Question

There is a whole collection of algebraically defined K-theories. My understanding is that algebraic K-theory is a presheaf of spectra $K$ on $\textbf{Sch}/S$ such that the homotopy groups of $K(X)$ give the algebraic K-groups of the $S$-scheme $X$.

What we can do is give $\textbf{Sch}/S$ a grothendieck topology, and find a fibrant replacement of $K$ in the local stable model structure on presheaves of spectra on $\textbf{Sch}/S$. These give us different K-theories. For example, if we take the etale topology (and take $K/n$) we get etale K-theory (with mod $n$ coefficients).

What is this process of finding a fibrant replacement geometrically? What is happening, and what does etale K-theory, say, measure?

Answer 1

Running off Denis' answer: the geometric content of Thomason's answer is surprising and he says this himself in the introduction to that legendary paper. Let me say some stuff that I've unpacked from his comments:

1. On the one hand, one can recover the intersection theory of algebraic varieties from $K$-theory: the Gersten resolutionin $K$-theory tells us that if $X$ is a regular $k$-scheme $X$, then $H^n_{Zar}(X; \mathcal{K}_n) \simeq CH^n(X)$. However, one consequence of Thomason's theorem is that $K/\ell^{\nu}[\beta^{-1}](X) \simeq K^{Top}/\ell^{\nu}(X)$ whenever $X$ is $\mathbb{C}$-scheme. This tells us that the process of Bott-inversion, which gives us etale descent, somehow reduces the subtleties of algebro-geometric intersection theory to the cruder topological case (someone more knowledgeable about the Hodge conjecture can perhaps comment on this).

2. If you believe that $K$-theory is about vector bundles, then the above theorem tells us (vaguely, I guess) that the "difference" between topological and algebraic vector bundles lie in $\beta$-torsion phenomenon. Since $\beta \in K^2(X, \ell^{\nu})$, the similarity between algebraic and topological vector bundles seems to lie only in $K_0, K_1$, i.e., they are the "same up to codimension one" (to paraphrase Thomason) by the formula $H^n_{Zar}(X; \mathcal{K}_n) \simeq CH^n(X)$ above

3. Just as Denis indicated, after Thomason's theorem, we get a spectral sequence that makes no reference to etale $K$-theory: its $E^2$-page is $H^p_{et}(X; \mathbb{Z}/\ell^{\nu}(q))$ and it converges to $K/\ell^{\nu}_{p-q}(X)[\beta^{-1}]$. Since the positive homotopy groups of etale $K$-theory and Bott-inverted $K$-theory are the same, it tells us that mod $\ell^{\nu}$-connective $K$-theory has a strong relationship with etale cohomology! This seems to be a general philosophy. For example, although $K$-theory is not $\mathbb{A}^1$-invariant on smooth schemes, Weibel proved that mod $\ell^{\nu}$-K-theory is (not saying that you can reproved this theorem using Thomason, however). This seems to indicate that, the fact that torsion etale cohomology is much better behaved than integral etale cohomology, is also shared by $K$-theory.

4. Lastly, I seem to be indicating that the distance between etale $K$-theory and usual $K$-theory is this Bott inversion. However, the recent work of Clausen, Mathew, Noel, Naumann tells us that this isn't really the case - the distance between etale $K$-theory and usual $K$-theory is covered by "telescopic localization" where Bott-inversion is just one case of telescopic localization.

Answer 2

Fibrant replacement is essentially sheavification with respect to the corresponding topology. So étale K-theory is nothing more than the part of K-theory that satisfies étale descent. Concretely (and computationally) this means that you have a spectral sequence (the descent spectral sequence)

$$ H^*(X_{ét},\pi_*K)\Rightarrow \pi_*K^{ét}(X) $$

(this is the analogue of Čech cohomology computations for sheaves of spectra) and similarly for the other topologies (except it turns out that algebraic K-theory is already a sheaf for the Nisnevich topology and not only for the Zariski topology: this is a consequence of the localization sequence).

One very cool fact, due to Thomason, is that étale sheavification for algebraic K-theory turns out to be the same thing as inverting a Bott element (at least after completing at a prime away from the characteristic of the base field and up to some connectiveness shenanigans).

Whether this is "geometric" or not I guess it depends on whether you are used to think of sheaf theory as geometric or not.