LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

General equicharacteristic scope and the standing filtration restriction

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

Statement

Assume the Axiom of Choice. Every nonempty finite-dimensional Noetherian equicharacteristic scheme in EM’s sense lies in the source filtration’s finite-valuative-dimension domain: its structure map factors through a prime field $F = Q$ or $F_{p}$ , it is qcqs, and its source valuative dimension equals its finite Krull dimension. Define its motivic filtration by the same source prime-field filtered algebra. On schemes over the standing field this is exactly the completed local filtration, including the actual stage maps to $K$ .

Facts & Assumptions

Given: The Axiom of Choice and a finite-dimensional Noetherian scheme $X$ with the stated global prime-field factorization, or its standing-field specialization.

Goal: Preserve the full equicharacteristic domain instead of silently restricting the theorem to one fixed base field.

[F1]

Elmanto–Morrow filtration input supplies the actual source filtered algebra for every prime field. Motivic filtration and image groups defines its standing-field version by actual restriction from that source.

[F2]

The local valuative convention matches the source convention and External valuative dimension comparison on locally Noetherian schemes give the source convention and equality for locally Noetherian schemes.

[F3]

The assumed The Axiom of Choice discharges the Choice premise of the selected filtration in Motivic filtration and image groups when its common-family perfect-complex realization and essentially small nonconnective-functor domain are invoked. No additional Choice step is needed for the geometric domain check.

Proof

1.1F1F2given

For nonempty $X$ , a Noetherian scheme has a finite affine cover. Every intersection of two affine opens is a Noetherian open and therefore quasi-compact; hence $X$ is qcqs. Its chosen prime-field factorization places it in the domain of [F1]. Since it is locally Noetherian, [F2] gives $\dim_{v} X = \dim X < \infty$ .

2.1F1F2F3step 1.1∎

Under the assumed Choice premise in [F3], evaluate the actual prime-field filtered algebra of [F1] on $X$ , with its source augmentation to stagezero $K (X)$ . If $X$ carries a structure over the standing field, [F1] says its locally defined filtration is precisely this source functor precomposed with forgetting to the prime field. Thus stages, transitions, multiplication and augmentation are the same restricted diagram. There is no independently selected filtration to compare, and no finite-type or fixed-field condition was inserted into step 1.1.

Depends on

Used by · 2

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