Eventual homotopy vanishing of the motivic tower

PDF p. 4: Motivic graded spectral sequence; PDF p. 9: Eventual filtration vanishing

Assume the Axiom of Choice. Let $X$ be a finite-dimensional Noetherian equicharacteristic scheme of Krull dimension $d$, with equicharacteristic meaning that its structure map factors through $\mathrm{Spec}\mathbb{Q}$ or $\mathrm{Spec}{\mathbb{F}}_p$. Use the actual source motivic filtration on that prime-field category; on schemes over the standing field this is exactly the completed local filtration. Then for each fixed integer $q\le 0$ used in the article there is ${\mathrm{\ell }}_0(q)$ such that ${\pi }_q{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{d+\mathrm{\ell }}K(X)=0(\mathrm{\ell }\ge {\mathrm{\ell }}_0(q))\mathrm{.}$ Indeed the same eventual vanishing holds in every fixed integer homotopy degree. The threshold may depend on that degree. The finite-valuative-dimension requirement of the source is satisfied because ${\dim }_{v}X=\dim X$ for Noetherian schemes. This is vanishing of the actual stage groups, stronger than merely vanishing of their images in $K_q(X)$.