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Exact arbitrary-field open-natural classical cycle interface

PDF p. 13: Proposition 6.1 support argument

Statement

Assume the Axiom of Choice and fixed site and universe bounds for the derived-category constructions. Let $r \in Z_{\geq 0}$ , let $k$ be perfect, and let $F$ be its prime subfield, either $Q$ or $F_{p}$ . Define $C_{r, F}$ to be the cdh sheafification of the left Kan extension of classical weight- $r$ motivic cohomology from smooth $F$ -schemes to qcqs $F$ -schemes. Restrict this functor to smooth separated finite-type $k$ -schemes and open immersions. There is an equivalence of these diagrams with $z^{r} (-, ∙) [- 2 r]$ , where homological degree $n$ is cochain degree $- n$ . Use the following actual route: the supplied canonical field-model equivalence identifies $C_{r, F}$ with the restriction of the absolute BEM cdh functor; its canonical comparison to the represented A¹ theory is an equivalence on these equicharacteristic schemes. Over the fixed $k$ , smooth slice pullback identifies that represented functor with classical motivic cohomology. Finally use the MVW Zariski-sheaf zigzag and Bloch descent. Every arrow or inverse equivalence is taken as a map of diagrams. Thus for an open immersion its cycle arrow is the actual open restriction.

External input

Import this restricted open-natural classical interface via the explicitly recorded zigzag. The base field k is fixed and perfect, with no finite-generation requirement over F. No separate construction of arbitrary nonflat cycle pullbacks is required. The completed field-model supplier states the canonical identification for any field B; its older application only used $B = Q$ , whereas here $B = F$ can also be $F_{p}$ . The new full source record explicitly retains that characteristic scope.

The global comparison is taken from the construction preceding BEM (6.4), supplemented by Levine's primary complex models, rather than inferred from (6.4) alone. Write $S m / / k$ for Levine's category with all smooth separated finite-type $k$ -schemes and smooth morphisms. His §6.5 supplies an isomorphism of derived sheaf objects on that whole category; BEM's proof of Theorem6.3 recalls its agreement in the homotopy category of presheaves. This is stronger than a natural isomorphism of cohomology groups, or a diagram in the homotopy category of the values. Only its restriction to open immersions and one fixed weight is used. The spectrum realization in Example8.2.2(3), together with the established classical slice identification recalled in BEM, supplies the represented-to-classical bridge. No compatibility between different weights or multiplicative comparison with cycles is claimed.

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.

[A1]

The Axiom of Choice is the existing ambient assumption for the imported derived-category models and their fixed small sites in steps 1.1–3.1. It is not an additional field or finite-generation hypothesis. The Axiom of Choice

[F1]

The canonical field-model equivalence supplied by External canonical field restriction identifies the BEM and EM cdh models, BEM Corollary1.11, and its represented-presheaf identification in Remark4.21.

[F2]

The global classical comparison constructions in BEM's proof of Theorem6.3 before (6.4), Levine §6.5 and Example8.2.2(3), and the established slice equivalences recalled in BEM Remark4.12 and that proof. Use their sheaf/complex models, not only their cohomology-group consequences. Levine §§1.2–1.4 distinguish the homotopy category of presheaves from presheaves of homotopy-category values.

[F3]

MVW Theorem16.7, Lemma19.4, Theorem19.8 and Proposition19.12 supply the classical-to-cycle sheaf zigzag, cycle homotopy invariance and descent. Definition19.13 and Example19.14 identify its flat maps, including open restriction.

Proof

1.1A1F1

Apply the canonical field-model equivalence to $B = F$ , and then restrict to $k$ -schemes. Compose it with the canonical cdh-to-A¹ comparison, an equivalence here by equicharacteristic. Remark4.21 identifies the resulting restriction to smooth $k$ -schemes with the presheaf represented by $s^{r} {K G L}_{k} [- 2 r]$ . All of these operations are on functors. This step uses neither finite generation of $k / F$ nor a choice of equivalence at each individual scheme.

2.1A1F2step 1.1

Use the established equivalences $s^{0} {K G L}_{k} ≃ s^{0} (1_{k}) ≃ H Z_{k}^{c l}$ recalled in [F2], where the last object is the classical motivic spectrum with Levine's complex models. The Bott/slice identification $s^{r} {K G L}_{k} ≃ T_{k}^{\otimes r} \otimes s^{0} {K G L}_{k}$ and the represented-presheaf functor produce the classical weight- $r$ functor on the entire smooth $k$ -category. Use the complex realization of [F2]: Levine §6.5 supplies the comparison on the whole category $S m / / k$ as a derived-sheaf isomorphism, and Example8.2.2(3) identifies the assembled spectrum with classical $H Z$ . Thus this is a comparison of the sheaf/presheaf objects before evaluation. The fixed-site statement (6.4) is a consequence of this construction, not a substitute for its global scope. The finite-field qualification causes no loss: the K-theory A3 hypothesis and the P¹-stable finite-field case are explicitly covered by the source contracts.

3.1F2F3step 2.1

The derived-sheaf and homotopy-presheaf comparisons just invoked can be represented by zigzags of maps between the corresponding models. Their isomorphisms become equivalences in the localized category of diagrams; they are not merely commuting squares after independently passing each value to its homotopy category. Restriction to the category of open immersions preserves those equivalences. Derived sections of the sheaf models and the source descent identifications supply the presheaf values. This is the coherence needed here: choose the global comparison once and then restrict it, rather than choosing unrelated comparisons on different small sites.

4.1F3step 3.1

Apply the actual MVW sheaf zigzag $Z (r) ≃ Z_{S F} (r) \to z^{r} (- \times A^{r}, ∙) [- 2 r] \leftarrow z^{r} (-, ∙) [- 2 r]$ . Its middle arrow is the cycle inclusion of Lemma19.4, a Zariski quasi-isomorphism by Theorem19.8. Its last arrow is projection flat pullback, a quasi-isomorphism by cycle homotopy invariance. Proposition19.12 identifies derived sections with cycle values. These are maps on the category of open immersions: the inclusion and projection pullback commute with restriction, and Example19.14 identifies the induced flat map with actual cycle pullback. They therefore give exactly the desired cycle arrows.

5.1F1F2F3step 1.1step 2.1step 3.1step 4.1∎

Compose the comparisons in steps 1.1–4.1 and invert only equivalences of the diagrams. An inverse of an equivalence in that functor category is again a map of diagrams, so the composite gives the asserted equivalence on all smooth separated finite-type $k$ -schemes and open immersions. No uniqueness, canonicity across weights, arbitrary nonflat cycle pullback or multiplicative compatibility is inferred. In particular every open arrow is the actual cycle restriction, with the declared homological-to-cochain indexing.

Local application provenance

The numbered argument is a locally assembled, AI-altered application of the imported comparison constructions. The literature-derived proof provenance records those external inputs; it does not attribute this five-step assembly verbatim to a published source. The existing source-reading records are retained from earlier authoring and adjudication. This consumer maintenance inspected the current item and its supplier contracts, without rereading the primary papers or independently judging their deep inputs.

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

Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes · Theorem7.5 and Remark7.9; Corollary1.11 and preceding canonical comparison(1.9); Remark3.41 maps(3.42)–(3.43); Remark4.12; Definitions4.15/4.17 and Remark4.18; Theorem1.1(8); Remark4.21; proof of Theorem6.3, paragraphs preceding (6.4); §3.2 mapping-presheaf construction; formula(4.7)

The absolute cdh functor restricts canonically over a field to the classical smooth-field cdh-local extension. Its canonical map to the A¹/slice theory is an equivalence on equicharacteristic schemes. Smooth pullback commutes with slice formation, and over a fixed field the represented smooth theory is classical motivic cohomology. These are natural functor constructions, not an arbitrary choice of objectwise equivalences.

Hypotheses: Every field B in the base-independence assertion; any qcqs equicharacteristic scheme for Corollary1.11; fixed perfect k for the classical smooth comparison used here; smooth separated finite-type k-schemes and open immersions.

Reading limits: Historical source-reading record retained from the prior draft: Read exact Theorem7.5 and short proof, Remark7.9, full Corollary1.11 statement and preceding explanation, full Remark3.41, Remark4.12 and Definitions4.15/4.17 with Remark4.18, Theorem1.1(8). The deep proof of the equicharacteristic comparison and classical slice identification is imported, not reconstructed. Served HTML title date24August2026 is retained despite v1 URL. Current consumer maintenance inspected the current item, its canonical field-model supplier and the parallel open-natural importer; it did not reread these primary papers or independently judge their deep inputs. Adjudication 527196b903d4 additionally read the served HTML title/version lines, Theorem1.1(8), Corollary1.11, the mapping-presheaf construction in §3.2, Remarks3.41, 4.12, 4.18, 4.21, formula(4.7), the opening comparison paragraphs and (6.4) with its qualifications in the proof of Theorem6.3, Theorem7.5 and Remark7.9. The deep slice identification is imported. The global presheaf assertion preceding (6.4), rather than that fixed-site display alone, is used below.

Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture notes on motivic cohomology · Theorem16.7 and Suslin–Friedlander construction p.126; Lecture19 pp.159–163: Lemma19.4, Theorem19.8, Proposition19.12, proof of Theorem19.1, Definition19.13 and Example19.14

The classical complex has the actual Zariski-sheaf comparison zigzag Z(r)≃Z_SF(r)→z^r(-×A^r,*)[-2r]. Bloch complexes satisfy Zariski descent, and projection pullback z^r(-,*)→z^r(-×A^r,*) is a quasi-isomorphism. For flat maps the comparison uses actual flat pullback, hence open restriction.

Hypotheses: Smooth separated schemes over a perfect field; r≥0. The required U and U\T are quasi-projective finite-type opens of an affine scheme.

Reading limits: Historical source-reading record retained from the prior draft: Read Theorem16.7 statement and its preceding construction, and all Lecture19 pp.159–163 including the displayed zigzag, quasi-isomorphism, descent proof, proof of19.1 and flat-pullback Example19.14. The deep Suslin comparison and moving/localization foundations invoked there are imported; no claim to reconstruct all prior lectures or general nonflat pullbacks. Current consumer maintenance inspected the current item, its canonical field-model supplier and the parallel open-natural importer; it did not reread these primary papers or independently judge their deep inputs. Adjudication 527196b903d4 read the primary PDF, printed p.126 (construction and Theorem16.7), and pp.159–163 (including the actual embedding, sheaf comparison, descent and flat-pullback description). This supplies the cycle maps and descent, not the preceding slice-to-classical bridge. Exact locally read PDF: 228 pages, creation metadata 18 July 2006, SHA256 bb07651d8cc1b38f2552204e6c777cc7570c19e35d779987907a44efc2669486; the URL has no immutable revision identifier.

Marc Levine, The homotopy coniveau tower (arXiv:math/0510334v1, 16 October 2005) · §§1.2–1.4, pp.7–8; §3.3, p.18; Theorem4.1.1, pp.20–21; Theorem6.4.1, p.35; §6.5, pp.37–38; Example8.2.2(3), pp.45–46; Theorem9.0.3 and Remark9.0.4, p.47; §§10.4–10.5, pp.59–61. These are v1 page and theorem numbers.

The Bloch complexes admit fibrant complexes of Nisnevich sheaves Z^r on Sm/k whose restriction is isomorphic to the cycle functor in the derived category of sheaves on Sm//k. The latter category has all smooth separated finite-type k-schemes as objects and only smooth morphisms. Their Eilenberg–MacLane realizations form the classical motivic spectrum HZ. The homotopy-coniveau and slice constructions identify its zeroth-slice realization; BEM explicitly recalls the resulting comparison with s^0 KGL and the global presheaf comparison before (6.4).

Hypotheses: Here k is perfect and r>=0; no finite-generation assumption over its prime field. Open immersions belong to Sm//k. Theorem4.1.1 requires axiom A3 for finite k; Theorem6.4.1 verifies it for K-theory, and Remark9.0.4 explains the finite-field scope of the P^1-stable comparison. No general infinite-field restriction from §7 is imposed on the stated P^1-stable result.

Reading limits: Read the cited definitions, theorem statements and construction paragraphs in the cached primary v1 PDF/text; visually checked printed pp.37 and45. Read the proof of Theorem6.4.1, the proof paragraph and map description in Example8.2.2(3), and the closing map identification on p.61. Did not reconstruct moving lemmas, the long reverse-cycle-map construction or the full slice-comparison proofs. Read the conditional qualification in Corollary10.5.2 and the opening of §11; no claim depending on its working Conjecture11.1.1 is imported. BEM supplies the established slice identification used here. The arXiv v1 numbering differs from the 2008 journal article cited by BEM; no direct reading of the journal version is claimed. Exact locally read v1 PDF SHA256: 5f268002b5ec36a74932a48ac051712c9c19c41a8b76e5bcb5dcdf0800538190.

Sources

Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes · Theorem7.5 and Remark7.9; Corollary1.11 and preceding canonical comparison(1.9); Remark3.41 maps(3.42)–(3.43); Remark4.12; Definitions4.15/4.17 and Remark4.18; Theorem1.1(8); Remark4.21; proof of Theorem6.3, paragraphs preceding (6.4); §3.2 mapping-presheaf construction; formula(4.7)
Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture notes on motivic cohomology · Theorem16.7 and Suslin–Friedlander construction p.126; Lecture19 pp.159–163: Lemma19.4, Theorem19.8, Proposition19.12, proof of Theorem19.1, Definition19.13 and Example19.14
Marc Levine, The homotopy coniveau tower (arXiv:math/0510334v1, 16 October 2005) · §§1.2–1.4, pp.7–8; §3.3, p.18; Theorem4.1.1, pp.20–21; Theorem6.4.1, p.35; §6.5, pp.37–38; Example8.2.2(3), pp.45–46; Theorem9.0.3 and Remark9.0.4, p.47; §§10.4–10.5, pp.59–61. These are v1 page and theorem numbers.