  The Quillen-Lichtenbaum conjecture for smooth complex varieties states that
algebraic and topological K-theory with finite coefficients become isomorphic
in high degrees. We define the "Quillen-Lichtenbaum dimension" of a variety in
terms of the point where this happens, show that it is surprisingly computable,
and analyze many examples. It gives an obstruction to rationality, but one that
turns out to be weaker than unramified cohomology and some related birational
invariants defined by Colliot-Th\'el\`ene and Voisin using Bloch-Ogus theory.
Because it is compatible with semi-orthogonal decompositions, however, it
allows us to prove some new cases of the integral Hodge conjecture using
homological projective duality, and to compute the higher algebraic K-theory of
the Kuznetsov components of the derived categories of some Fano varieties.
