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Fixed-site classical cycle comparison on every smooth ambient

PDF p. 13: Proposition 6.1 support argument

Statement

Assume the Axiom of Choice and fixed universe and site bounds. Let $k$ be perfect, let $F$ be its prime subfield, and let $r \in Z_{\geq 0}$ . Let $C_{r, F}$ be the cdh-local left Kan extension of classical weight- $r$ motivic cohomology from smooth $F$ -schemes to qcqs $F$ -schemes. Fix a smooth separated finite-type $k$ -scheme $S$ . On the finite-type separated objects of the small étale site $S_{e t}$ , there is an equivalence of presheaves with values in $D (Z)$ $C_{r, F} ∣_{S_{e t}} ≃ z^{r} (-, ∙) [- 2 r] .$ In particular, the restriction to the entire diagram of opens of this fixed $S$ is coherent, and each cycle arrow is actual open restriction. On that open diagram it also identifies $C_{r, F}$ with the derived-section functor of the MVW classical coefficient complex $Z (r)$ . Homological degree $n$ is cochain degree $- n$ before shifting. The equivalence may depend on $S$ and $r$ ; no compatibility between choices for different ambients or weights, or multiplicative compatibility, is asserted. There is no finite-generation requirement on $k / F$ .

External input

Import BEM's explicit equivalence (6.4) on each fixed smooth ambient's étale site, rather than infer a global diagram equivalence from objectwise comparisons.
Import its canonical field-model identification and equal-characteristic comparison, and the MVW sheaf-complex maps and descent on the fixed Zariski site. The deep source theorems are not reconstructed here.

The imported comparison results and their proofs are literature-derived. The numbered proof below is a locally authored application argument: it composes the supplied comparisons, restricts them to the fixed open diagram, and inverts equivalences in the corresponding functor categories. The separate local_argument_provenance: ai-altered records this local composition; the literature-derived proof provenance records the imported results and does not attribute the application argument as written to either source.

Facts & Assumptions

Given: The hypotheses of the statement.

Goal: Compare the whole fixed-site diagram before taking cohomology or fibers.

[F1]

External canonical field restriction identifies the BEM and EM cdh models supplies the canonical field-model identification for every field $F$ ; BEM Corollary 1.11 supplies the natural equivalence from that absolute cdh functor to its A¹ theory on equal-characteristic schemes.

[F2]

BEM Theorem 6.3 proof, equation (6.4), supplies an equivalence of presheaves of complexes on the fixed $S_{e t}$ in each nonnegative weight.

[F3]

MVW Theorems 16.7 and 19.8, Proposition 19.12 and Example 19.14 supply the actual fixed-site chain maps and descent recorded in the source reference.

[A1]

The Axiom of Choice and the fixed bounds are the ambient foundational convention for the imported derived constructions.

Proof

1.1A1F1F2

Restrict the canonical field-model equivalence and canonical equal-characteristic comparison to $S_{e t}$ . Every object here is an equal-characteristic qcqs scheme. Compose with the equivalence of presheaves of complexes (6.4) on this same fixed site. This gives the first assertion as an equivalence in the functor category, including all étale restriction maps; it is not an inference from separate objectwise equivalences.

1.2A1F3

Restrict to $Open (S)$ . On the small Zariski site use the actual zigzag $Z (r) ≃ Z_{S F} (r) \to z^{r} (- \times A^{r}, ∙) [- 2 r] \leftarrow z^{r} (-, ∙) [- 2 r]$ . Apply derived sections functorially to its sheaf quasi-isomorphisms. Proposition 19.12 identifies cycle values with their derived sections, and projection flat pullback is a quasi-isomorphism by cycle homotopy invariance. The chain maps commute with open restriction, so this identifies the MVW coefficient functor on all opens of $S$ with the cycle functor on that same diagram.

2.1step 1.1step 1.2∎

Invert these equivalences only in their fixed-site functor categories and compose them. Inverses of natural equivalences are natural equivalences, and restriction to any open subdiagram preserves them. This proves the classical-intermediate and actual-open-arrow assertions without choosing unrelated inverses at individual objects or requiring compatibility with another ambient.

Depends on

Used by · 2

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

Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, served v1 HTML · Theorem 7.5, Definition 7.8, Remark 7.9; comparison (1.9), Corollary 1.11; proof of Theorem 6.3, paragraphs preceding and following (6.4)

For every fixed smooth variety S over a field and r>=0, (6.4) is an equivalence of presheaves of complexes on S_et between the represented A¹ theory and the actual Bloch cycle complex shifted by -2r. The canonical cdh-to-A¹ comparison is an equivalence on equal-characteristic schemes. Remark 7.9 identifies the restriction of the absolute cdh functor with the cdh-local smooth-field left Kan extension.

Hypotheses: Fixed smooth separated finite-type ambient S over a perfect field k; r>=0; finite-type separated objects of its small étale site. The field-model assertion applies to every field, including the prime field F. No finite-generation assumption on k/F.

Reading limits: Current targeted reading of the served arXiv HTML: comparison (1.9), Corollary 1.11; Remark 4.21 and Definitions 4.15/4.17; the proof of Theorem 6.3 through (6.4) and its fixed-site qualification; Definition 7.1, Proposition 7.4 and its proof, Theorem 7.5 and its short proof, Definition 7.8 and Remark 7.9. The page identifies itself as arXiv:2508.09915v1, 13 August 2025, but bears title date August 24, 2026. No immutable August 2025 snapshot was verified. The deep results are imported; no reconstruction of slice, moving, base-independence or equicharacteristic comparison proofs is claimed.

Mazza–Voevodsky–Weibel, Lecture notes on motivic cohomology, Rutgers served PDF · Printed p.126, Theorem 16.7; printed pp.159–163, Lemma 19.4, Theorem 19.8, Proposition 19.12, Definition 19.13 and Example 19.14 (PDF pp.138–139 and 171–175)

On a fixed smooth scheme, the classical coefficient complex has a Zariski-sheaf zigzag through the Suslin–Friedlander complex to z^r(-×A^r,*)[-2r]. Cycle complexes satisfy Zariski descent, and projection flat pullback is a quasi-isomorphism. These chain maps commute with actual open restriction.

Hypotheses: Smooth separated schemes over a perfect field; r>=0; fixed small Zariski site. Applied to finite-type opens of S.

Reading limits: Current targeted reading of the served 228-page Rutgers PDF: printed p.126 Suslin–Friedlander construction and Theorem 16.7, with its short proof on p.127; printed pp.159–163, Theorem 19.1, Lemma 19.4, the small-site sheaf construction, Theorem 19.8 and its proof, Proposition 19.12 and its proof, proof of 19.1, Definition 19.13 and Example 19.14. Printed pp.126–127 are PDF pages 138–139 and pp.159–163 are PDF pages 171–175, using one-based PDF numbering. The earlier Suslin comparison and cycle homotopy-invariance/localization foundations are imported, not reconstructed. Only open restriction, not a general coherent nonflat cycle pullback, is used.

Sources

Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, served v1 HTML · Theorem 7.5, Definition 7.8, Remark 7.9; comparison (1.9), Corollary 1.11; proof of Theorem 6.3, paragraphs preceding and following (6.4)
Mazza–Voevodsky–Weibel, Lecture notes on motivic cohomology, Rutgers served PDF · Printed p.126, Theorem 16.7; printed pp.159–163, Lemma 19.4, Theorem 19.8, Proposition 19.12, Definition 19.13 and Example 19.14 (PDF pp.138–139 and 171–175)