On the negative K-theory of schemes in finite characteristic

We show that if X is a d-dimensional scheme of finite type over an infinite perfect field k of characteristic p>0, then Ki(X)=0 and X is Ki-regular for i<−d−2 whenever the resolution of singularities holds over k. This proves the K-dimension conjecture of Weibel [C. Weibel, K-theory and analytic isomorphisms, Invent. Math. 61 (1980) 177–197, 2.9] (except for −d−1⩽i⩽−d−2) in all characteristics, assuming the resolution of singularities.