External geometric properties of the actual motivic comparison fiber

PDF p. 4: Motivic dimension vanishing

For a prime field $F$ and $j\ge 0$, define the actual comparison fiber $F_{j,F}=\mathrm{fib}(M_{j,F}\stackrel{{\alpha }_{j,F}}{\to }{C}_{j,F})\mathrm{.}$ It is a finitary Nisnevich sheaf, satisfies $L_{\mathrm{c}\mathrm{d}\mathrm{h}}{F}_{j,F}=0$ and pro-cdh descent on Noetherian $F$-schemes, and for every $F$-algebra $A$ and finitely generated nilpotent ideal $I\subseteq A$ one has $\mathrm{fib}(F_{j,F}(A)\to F_{j,F}(A\mathrm{/}I))\in D(\mathbb{Z}{)}^{\le j}\mathrm{.}$

Given: The actual complexes and comparison of External prime-field cdh comparison and cdh dimension bound, with prime field $F$ and nonnegative weight.

Goal: Import exactly the geometric hypotheses later normalized for the dimension criterion.

[F1] EM Proposition8.12, with its proof, gives all the stated properties of this specified fiber; Theorem8.2 is its pro-cdh ingredient.

Import [F1] under the recorded hypotheses and reading limits. Throughout this package and its dimension-criterion application, pro-cdh descent means the explicit degreewise convention in EM Theorem 8.2, footnote 14: every cohomology group of the levelwise total fiber for an abstract blowup square and its infinitesimal thickenings is zero as a pro abelian group. The source's additional comparison with cartesianness in the unbounded category $\mathrm{Pro}(D(\mathbb{Z}))$ is not imported. The dimension criterion's proof needs only this degreewise condition. In particular, no degreewise extraction from vanishing K-groups is used. The characteristic-zero derived de Rham and characteristic-$p$ syntomic-rigidity arguments underlying the relative bound, and the pro-cdh machinery, remain explicit imported foundations. The local argument below checks their application and the shift required by the criterion instead of claiming to reprove them.