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Exact natural regular motivic-to-cdh comparison

PDF p. 13: Proposition 6.1 support argument

Statement

For a prime field $F$ and $r \geq 0$ , the actual source comparison $η_{r} : Z (r)_{m o t, F} \to C_{r, F} = Z (r)_{c d h, F}$ is a natural transformation on all qcqs $F$ -schemes and is an equivalence on every regular Noetherian $F$ -scheme. Therefore on smooth finite-type schemes over any perfect field $k \supset F$ , it is an equivalence of diagrams for open restrictions, even when those schemes are not finite type over $F$ .

External input

Import the exact natural transformation on all qcqs prime-field schemes and its regular Noetherian equivalence. This supplies naturality before taking groups or fibers. It does not infer naturality from the pointwise wording of Theorem1.1(9). The smooth arbitrary-field cycle comparison is supplied independently by the next input.

Facts & Assumptions

Given: The hypotheses in the statement, with the Axiom of Choice and fixed universe bounds.

Goal: Construct the stated actual maps and prove their consequence.

[F1]

The exact source records in External input.

Proof

1.1F1

Import the actual transformation (6.1) and Theorem6.1(2).

2.1F1step 1.1∎

Its hypotheses are regular Noetherian, not smooth finite type over the prime field. Precompose this transformation with the forgetful functor on $k$ -schemes; regularity and Noetherianness of the underlying scheme are unchanged. Thus it applies to the stated smooth schemes and their open restrictions. The substantial regular comparison proof is imported, not reconstructed.

Depends on

Used by · 1

A coherent comparison of the particular open restriction

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.1(9) p.4; Construction4.34 p.31; §6 equation(6.1) and Theorem6.1(2) p.44; Corollary6.4 and footnote12 p.45

The natural transformation (6.1) exists on all qcqs prime-field schemes and is an equivalence on regular Noetherian schemes. Construction4.34 is prime-field smooth; footnote12 explains why the resulting smooth theory also agrees with classical motivic cohomology over any field.

Hypotheses: F=Q or F_p; r≥0; regular Noetherian underlying scheme for the equivalence. No finite-type-over-F hypothesis in Theorem6.1(2).

Reading limits: Read exact displayed map, theorem, corollary and complete short corollary proof and arbitrary-field footnote. Read Construction4.34 fully. Read the §6 opening argument and its main proof outline, not the entire Gersten-injectivity reconstruction. The natural regular comparison is imported; no claim that the prime-field smooth construction itself already covers every k-variety.

Sources

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.1(9) p.4; Construction4.34 p.31; §6 equation(6.1) and Theorem6.1(2) p.44; Corollary6.4 and footnote12 p.45