TheoremStatement: AI-adaptedProof: Literature-sourced

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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External actual motivic stagewise boundedness

PDF p. 4: Motivic graded spectral sequence; PDF p. 9: Eventual filtration vanishing

Statement

Assume the Axiom of Choice and the fixed universe bounds of the selected perfect-K-theory and motivic-filtration constructions. For a qcqs scheme $X$ over a prime field $F = Q$ or $F_{p}$ with finite valuative dimension, there exists an integer $B (X) \geq 0$ such that the actual EM filtration satisfies $F_{m o t}^{r} K (X) \in {S p}_{\geq r - B (X)}$ for every integer stage $r$ under the source’s constant extension to $r \leq 0$ . Equivalently its cohomological support, with $H^{m} (E) = π_{- m} E$ , is in degrees $\leq B (X) - r$ . The same actual tower is nonnegative, augmented by $F^{0} = K$ , multiplicative, and induces the source’s bounded multiplicative spectral sequence. The value of $B$ need not equal a Krull or valuative dimension.

Facts & Assumptions

Given: The Axiom of Choice, the fixed universe bounds, a qcqs scheme $X$ over the specified prime field with finite valuative dimension, the actual source filtration, and the source valuative convention.

Goal: Import the precise stagewise, rather than merely graded or total- $K$ , boundedness statement.

[A1]

The Axiom of Choice in The Axiom of Choice is inherited through Motivic filtration and image groups. It is used to invoke the selected common-family perfect- $K$ -theory realization when applying the source theorem to that local tower.

[F1]

EM Theorems 4.12(1) and 4.27(1), in the exact version and at the locators recorded above, assert boundedness for the stated qcqs schemes of finite valuative dimension. Section 2.2 defines this by a common cohomological upper bound $B - r$ on the individual stages for all integer indices.

[F2]

Elmanto–Morrow filtration input supplies the source tower, its multiplication and zeroth-stage equivalence. Motivic filtration and image groups identifies the local tower and its constant nonpositive extension. Valuative dimension and degreewise bounded towers fixes the valuative convention.

External input

Import [F1] with the full source records and reading limits. This actual bound is additional input beyond the completed existence-only supplier. The printed proof paragraphs contain false homological sign/type inequalities recorded in the assistance findings; those estimates are not used. Only the explicit theorem assertion interpreted by its definition, with some finite $B (X)$ , is imported. No deduction of stagewise vanishing from total K-theory vanishing is made.

Proof

1.1A1F1F2given

Under [A1], use the actual tower of [F2]. Apply Theorem 4.12(1) in characteristic zero or Theorem 4.27(1) in characteristic $p$ , as specified in [F1]. The hypotheses are exactly that $X$ is qcqs over the selected prime field and has finite valuative dimension. Import the qualitative boundedness assertion with its section 2.2 definition, giving a finite integer $B (X) \geq 0$ with cohomological support in degrees $\leq B (X) - r$ for every integer $r$ , including the constant nonpositive stages. This applies the external theorem without reconstructing its proof.

2.1F1F2step 1.1∎

Since $H^{m} (E) = π_{- m} E$ , step 1.1 is equivalent to $π_{i} F_{m o t}^{r} K (X) = 0$ for $i < r - B (X)$ , which is the displayed connectivity assertion. The indexing, augmentation and multiplication are the data of [F2]; the bounded multiplicative spectral sequence is included in [F1]. No particular dimension formula for $B (X)$ is claimed, and the erroneous numerical estimates in the printed proofs are not 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

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §2.2 pp.11–12; Theorem4.12(1) p.22 and Theorem4.27(1) p.29; Theorem1.1(1) p.3

For a qcqs scheme over Q or F_p of finite valuative dimension, the actual multiplicative motivic filtration is bounded in the explicit §2.2 sense: there exists an integer B≥0 such that stage r is supported in cohomological degrees≤B−r, equivalently homotopical degrees≥r−B. It is N-indexed and stage0 is K. Boundedness yields completeness and a bounded multiplicative spectral sequence.

Hypotheses: Qcqs over the selected prime field; finite valuative dimension in the Jaffard/EHIK convention. Noetherianity is not required for this statement. No particular numerical value B=dim_vX+c is imported.

Reading limits: This adjudication read the arXiv:2309.08463v1 PDF text: Theorem 1.1(1), printed p.3; all of section 2.2, pp.11–12; Theorems 4.12 and 4.27 and their full printed proofs, pp.22–23 and29. The PDF identifies v1 (15 September 2023), with title date 18 September 2023. New screenshots of the two proof pages failed with cache-miss errors, so no fresh page-image inspection is claimed. The false numerical proof estimates are retained as source findings and are not premises. The stated qualitative theorem is imported with its definition; trace/descent foundations and optimal bounds were not reconstructed.

Sources

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §2.2 pp.11–12; Theorem4.12(1) p.22 and Theorem4.27(1) p.29; Theorem1.1(1) p.3