External dimension criterion for a cdh-acyclic Nisnevich sheaf

PDF p. 4: Motivic dimension vanishing

Let $R$ be a base ring and $W:({\mathrm{S}\mathrm{c}\mathrm{h}}_R^{\mathrm{q}\mathrm{c}\mathrm{q}\mathrm{s}}{)}^{op}\to D(\mathbb{Z})$ a finitary Nisnevich sheaf. Assume:

1. $L_{\mathrm{c}\mathrm{d}\mathrm{h}}W\simeq 0$.
2. $W$ satisfies pro-cdh descent on Noetherian $R$-schemes.
3. For every Noetherian local henselian $R$-algebra $A$ and every nilpotent ideal $I$, $\mathrm{fib}(W(A)\to W(A\mathrm{/}I))\in D(\mathbb{Z}{)}^{\le 0}$.

Then $W(X)\in D(\mathbb{Z}{)}^{\le \dim X}$ for every Noetherian $R$-scheme $X$ of finite dimension.

Given: All three assumptions and the stated sheaf, continuity and scheme hypotheses.

Goal: Import the established geometric dimension criterion, including its exact input class.

[F1] EM Proposition8.10 is this statement with arbitrary base ring.

Import [F1] with the recorded reading limits. Finitary means that inverse limits of qcqs schemes with affine transition maps are sent to the corresponding filtered colimits of complexes. Nisnevich descent means derived descent for the étale completely decomposed covering families. The cdh topology is generated by Nisnevich covers and abstract blowup covers. An abstract blowup square consists of a proper finitely presented map $X^{\mathrm{\prime }}\to X$, a finitely presented closed immersion $Y\to X$, its pullback $Y^{\mathrm{\prime }}$, and an isomorphism $X^{\mathrm{\prime }}\setminus Y^{\mathrm{\prime }}\simeq X\setminus Y$. Pro-cdh descent here has exactly the degreewise convention of EM Theorem 8.2, footnote 14: for each abstract blowup square and its powers-of-ideal thickenings, every cohomology group of the levelwise total fiber is zero as a pro abelian group. No equivalence with a cartesian square in the unbounded category $\mathrm{Pro}(D(\mathbb{Z}))$ is used. In the modification step of Proposition 8.10, this degreewise hypothesis suffices: for fixed $n>\dim X$, the induction makes the two lower-dimensional relative cohomology systems vanish in the relevant degrees, and the pro long exact sequence gives the required isomorphism on degree $n$ for the two constant top terms. A nilpotent ideal satisfies $I^n=0$ for some $n$; a henselian local algebra has the usual unique lifting of simple roots from its residue field. These are hypotheses of the imported theorem, not new claims that a general sheaf has them.