External prime-field cdh comparison and cdh dimension bound

PDF p. 4: Motivic dimension vanishing

Fix a prime field $F$ and $j\ge 0$. The actual EM functor $M_{j,F}=\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}$ on qcqs $F$-schemes has a natural comparison ${\alpha }_{j,F}:M_{j,F}\to C_{j,F}$, where $C_{j,F}=\mathbb{Z}(j{)}_{\mathrm{c}\mathrm{d}\mathrm{h}}=L_{\mathrm{c}\mathrm{d}\mathrm{h}}{L}_{\mathrm{K}\mathrm{a}\mathrm{n}}\mathbb{Z}(j{)}_{\mathrm{c}\mathrm{l}\mathrm{a}}\mathrm{.}$ Here $L_{\mathrm{K}\mathrm{a}\mathrm{n}}$ extends the classical smooth-$F$ motivic complex to qcqs $F$-schemes and $L_{\mathrm{c}\mathrm{d}\mathrm{h}}$ is derived cdh sheafification. The comparison identifies $C_{j,F}$ with $L_{\mathrm{c}\mathrm{d}\mathrm{h}}{M}_{j,F}$. For every Noetherian $F$-scheme $X$ of finite dimension $d$, $H^i(C_{j,F}(X))=0$ for $i>j+d$.

Given: The prime-field motivic sheaf already supplied by Elmanto–Morrow motivic complex input, with cohomology in Cohomology descends to the derived category.

Goal: Specify the actual comparison target and import its precise outer-term dimension bound.

[F1] EM comparison map(1.2), Theorem1.3(1), and definition(3.5) give the actual comparison and cdh target, on the same prime-field site as the supplied motivic complex.

[F2] The p.56 paragraph following Remark8.8 explicitly states $C_{j,F}(X)\in D(\mathbb{Z}{)}^{\le j+d}$ in the Noetherian dimension-$d$ context, invoking Theorem3.5(1).

Import [F1]–[F2] with the full source records and reading limits. A complex belongs to $D(\mathbb{Z}{)}^{\le N}$ when its cohomology vanishes in every degree $>N$. The source obtains the cdh-motivic bound from bounds on the individual filtered stages and shifted graded terms of KH, rather than from vanishing of the total KH-groups alone. The construction and comparison are imported established inputs, not local reconstructions of cdh motivic cohomology.