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Elmanto–Morrow smooth comparison input

PDF p. 4: Property (4)

Statement

Let $k$ be a field, $X$ a smooth finite-type $k$ -scheme, and $j \geq 0$ . The Elmanto–Morrow complex $Z (j)_{m o t} (X)$ is equivalent in the derived category of abelian groups to Bloch's cycle complex, regarded cohomologically and shifted by $- 2 j$ . It consequently represents the classical Voevodsky weight- $j$ motivic cohomology of $X$ .

For the filtration comparison in the selected local perfect- $K$ -theory realization, assume the Axiom of Choice and its fixed suitable universes. Under these assumptions the Elmanto–Morrow filtration of $K (X)$ is naturally equivalent as a multiplicative filtered spectrum to the filtration arising from Voevodsky's slice filtration.

The classical motivic complex and the slice filtration are the constructions recalled in the cited source §3.1. A multiplicative filtered equivalence is an equivalence of towers compatible with their transition maps and multiplication.

External input

This is the cited theorem's smooth comparison, together with the identification recalled in §3.1. It is imported from the literature; its geometric proof and motivic-homotopy constructions are not locally reconstructed. The restriction to finite type matches the locally defined cycle complex.

Facts & Assumptions

Given: A field $k$ , a smooth finite-type $k$ -scheme $X$ , and $j \geq 0$ ; for the filtration assertion in the selected local realization, the Axiom of Choice and its fixed suitable universes.

Goal: Supply the complex and filtered-spectrum comparisons under those hypotheses.

[F1]

Elmanto–Morrow Theorem 1.1(9) states the smooth complex and multiplicative filtered-spectrum equivalences; §3.1 identifies the shifted cycle complex with the classical motivic complex.

[A1]

The assumed The Axiom of Choice and fixed universes discharge the premise of Motivic filtration and image groups when its selected common-family perfect-complex realization is invoked for the filtered comparison in step 2.1. The complex and classical-cohomology identifications of Motivic complexes and cohomology and Bloch cycle complexes and higher Chow groups in step 1.1 use no such local perfect- $K$ realization. This records the local supplier contract, without asserting foundational necessity for the published theorem.

Proof

1.1givenF1

$X$ is quasi-compact and quasi-separated because it is finite type over a field, and it is smooth by hypothesis. Apply the complex comparison of [F1] in nonnegative weight. With the specified cycle and motivic-complex conventions this gives the derived equivalence and its classical interpretation. These are imported assertions, with no local reconstruction of their geometric proof.

2.1A1F1step 1.1∎

For the filtration assertion, invoke the selected local tower under [A1]. The filtered comparison in [F1] then identifies this tower with the classical slice tower as a multiplicative filtered spectrum. Together with step 1.1 this proves the assertions with their respective premises; no numerical connectivity estimate from §3.1 is used.

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

Motivic cohomology of equicharacteristic schemes · Theorem 1.1(9), PDF page 4 of 74; §3.1, equation (3.1) and slice-filtration discussion, PDF pages 12–13 of 74

For a smooth scheme X over a field and j ≥ 0, the Elmanto–Morrow weight-j complex is equivalent to Bloch’s cycle complex shifted by −2j. Its motivic K-theory filtration is naturally equivalent, as a multiplicative filtered spectrum, to the filtration from Voevodsky’s slice filtration. Section 3.1 identifies the shifted cycle complex with classical motivic cohomology.

Hypotheses: X is quasi-compact and quasi-separated and smooth over a field; j is a nonnegative integer. The application below uses smooth finite-type schemes.

Reading limits: The previous record reported reading the theorem and §3.1 in arXiv HTML. This maintenance read the exact v1 PDF extracted text: title/version (15 September 2023; title date 18 September 2023), Theorem 1.1(9), and §3.1 through the classical slice-filtration discussion. The HTML endpoint returned a different title date (24 August 2026), so version binding here uses the PDF. Screenshot requests for PDF pages 4 and 12 failed with cache-miss errors; no fresh image inspection is claimed. The source proofs and motivic-homotopy foundations were not reconstructed. The numerical connectivity display in §3.1 is not imported.

Sources

Motivic cohomology of equicharacteristic schemes · Theorem 1.1(9), PDF page 4 of 74; §3.1, equation (3.1) and slice-filtration discussion, PDF pages 12–13 of 74