Rational splitting of the motivic tower

Assume the Axiom of Choice and use the universes of the supplied perfect-complex $K$-theory interface. Let $X$ be a quasi-compact quasi-separated scheme over $\mathbb{Q}$ or over ${\mathbb{F}}_p$ for a prime $p$, of finite valuative dimension in the supplied convention. Let $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{\bullet }K(X)$ be the Elmanto–Morrow motivic filtration on that nonconnective $K$-theory spectrum. Rationalization means smashing each stage with the rational sphere spectrum. There is a natural equivalence of augmented filtered spectra from this rationalized tower to the tail tower

$q⟼{⨁}_{j\ge q}\mathbb{Q}(j{)}^{\mathrm{m}\mathrm{o}\mathrm{t}}(X)[2j](q\ge 0),$

extended constantly for $q\le 0$. Here $\mathbb{Q}(j{)}^{\mathrm{m}\mathrm{o}\mathrm{t}}(X)$ denotes the rationalization of Elmanto–Morrow's weight-$j$ motivic cohomology spectrum, and $[2j]$ is its $2j$-fold suspension. In particular, the associated motivic spectral sequence has zero differentials after rationalization. The induced image filtration has successive quotients given by the homotopy groups of the corresponding rational weight pieces.

Valuative dimension and degreewise bounded towers

Nonconnective K theory of perfect complexes

The existence, boundedness and natural rational filtered splitting of the Elmanto–Morrow tower are imported from the cited theorems. Their construction and proof are not reconstructed here. In characteristic zero the cited theorem additionally asserts multiplicativity of its equivalence; no multiplicativity of the splitting is asserted here in positive characteristic. The splitting by itself is not a terminal-edge isomorphism: a particular edge comparison also requires the appropriate graded identification and high-filtration vanishing.