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Prime-field evaluation and compatibility with the standing theory

PDF p. 4: Motivic dimension vanishing

Statement

For any prime field $F \in {Q, F_{p}}$ and $j \geq 0$ , define the general EM groups on qcqs $F$ -schemes by ${\hat{H}}_{F}^{i, j} (X, Z) = H^{i} (M_{j, F} (X)) .$ For a field $k$ with prime subfield $F$ , restriction of this interface to qcqs $k$ -schemes is exactly the standing $H_{m o t}^{i, j}$ functor, including its pullbacks. Thus the same notation can be used for the general prime-field evaluation and its fixed-field restriction. Write $F_{j, F}$ for the actual fiber of the natural map $M_{j, F} \to C_{j, F}$ , with fiber sequence $F_{j, F} (X) \to M_{j, F} (X) \to C_{j, F} (X)$ at every $X$ .

Facts & Assumptions

Given: The completed prime-field motivic sheaves, the standing fixed-field definition and their actual comparison to cdh motivic complexes.

Goal: Define the larger-scope interface without substituting a different motivic theory.

[F1]

Elmanto–Morrow motivic complex input supplies $M_{j, F}$ for every prime field and explicitly obtains the $k$ -scheme functor by restriction along forgetting $k$ to $F$ .

[F2]

Motivic complexes and cohomology defines its groups by objectwise cohomology of exactly that restricted functor; Cohomology descends to the derived category supplies functoriality on $D (Z)$ .

[F3]

External prime-field cdh comparison and cdh dimension bound supplies the actual natural map $α_{j, F}$ to the specified cdh complex.

Proof

1.1F1F2

Apply cohomology to the actual prime-field functor in [F1]. By [F2] this defines the displayed general groups and their pullbacks. On a qcqs $k$ -scheme, forgetting its $k$ -structure before evaluation gives precisely the complex used by the standing definition, by its stated construction. The underlying complexes and maps are thus the same restricted diagram, so the groups and pullbacks agree without an independently chosen comparison isomorphism.

2.1F1F2F3step 1.1∎

Take the pointwise derived fiber of [F3]. Limits of functor diagrams are computed objectwise, so it gives an actual presheaf and the stated natural fiber sequence at every $X$ . On restriction to $k$ -schemes this is the same sequence, again by precomposition. Empty schemes give zero complexes by empty-cover Nisnevich descent. The weight range is exactly $j \geq 0$ as in [F1].

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