A K-theory vanishing result, which was previously a conjecture of Weibel, has been proved by Guillermo Cortinas, Christian Haesemeyer, Marco Schlichting, and Charles A. Weibel. The result will appear in the paper “Cyclic homology, cdh-cohomology and negative K-theory” in the Annals of Mathematics.
Abstract
In this paper, we use properties of the Chern character from K-theory to negative cyclic homology to prove a conjecture of Weibel — asserting that the K-theory of a scheme vanishes in degrees less than minus the dimension of the scheme — for schemes essentially of finite type over a field of characteristic 0. To do so, we prove a blow-up formula for various versions of cyclic homology, we show that Cortinas' infinitesimal K-theory satisfies cdh-descent, and we show that Zariski-cohomology surjects onto cdh-cohomology of O_X in degree d=dim X, X a scheme essentially of finite type over a field of characteristic 0.