Étale K-theory

Summary

Étale K-theory is a version of algebraic K-theory with finite coefficients that is built to satisfy descent for the étale topology. It is obtained by gluing the mod $\ell$ K-theory of the étale neighbourhoods of a scheme. So it is computed by étale cohomology through a descent spectral sequence. Algebraic K-theory maps to it, and the Quillen–Lichtenbaum conjecture, now a theorem, says this map is an isomorphism in high degrees. It goes back to Friedlander and to Dwyer and Friedlander, and Thomason related it to algebraic K-theory with the Bott element inverted.