Étale K-theory

Remarks

There is a natural comparison map $K(X)/\ell \to K^{\mathrm{et}}(X; \Z/\ell)$. Assume that the étale cohomological dimension of $X$ for $\ell$-torsion sheaves is finite. Then a descent spectral sequence computes $K^{\mathrm{et}}_*(X; \Z/\ell)$ from the étale cohomology of $X$ with coefficients in the sheaves associated to $U \mapsto K_q(U; \Z/\ell)$.

Example: let $X = \operatorname{Spec} k$ with $k$ an algebraically closed field in which $\ell$ is invertible. Every étale $k$-algebra is a finite product of copies of $k$, and K-theory turns finite products into products, so the presheaf already satisfies étale descent. Hence $K^{\mathrm{et}}(k; \Z/\ell) = K(k)/\ell$, and $K^{\mathrm{et}}_n(k; \Z/\ell) \cong \Z/\ell$ for even $n \ge 0$ and $0$ otherwise, by Suslin's computation.

Non-example: the construction is not used when $\ell$ is divisible by the characteristic of a residue field of $X$. For $\ell = p$ in characteristic $p$, the étale cohomology of $\Z/p$ sheaves behaves differently. The comparison with K-theory then takes another form.

The original constructions, by Friedlander and by Dwyer and Friedlander, used the étale homotopy type of $X$ instead of hypercohomology.