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.

The global dimension bound for the specified comparison fiber

PDF p. 4: Motivic dimension vanishing

Statement

For every prime field $F$ , every Noetherian $F$ -scheme $X$ of finite dimension $d$ , and every $j \geq 0$ , the actual comparison fiber satisfies $F_{j, F} (X) \in D (Z)^{\leq j + d} .$ Equivalently $H^{i} (F_{j, F} (X)) = 0$ for $i > j + d$ .

Facts & Assumptions

Given: $F, X, d, j$ as stated and the actual fiber rather than an abstract object with matching groups.

Goal: Apply the geometric criterion and undo the normalization with the correct cochain sign.

[F1]

Normalizing the comparison fiber for the dimension criterion checks all the criterion hypotheses for $W_{j, F} = F_{j, F} [j]$ .

[F2]

External dimension criterion for a cdh-acyclic Nisnevich sheaf applies to every finite-dimensional Noetherian scheme over the chosen base ring. Prime-field evaluation and compatibility with the standing theory specifies the pointwise fiber.

Proof

1.1F1F2given

Take the base ring in [F2] to be $F$ . A Noetherian scheme is quasi-compact and quasi-separated: it has a finite affine cover, and the intersection of two affine opens is a Noetherian open, hence quasi-compact. Thus $X$ belongs to the criterion’s scheme domain. With all sheaf and relative hypotheses supplied by [F1], the criterion gives $W_{j, F} (X) \in D (Z)^{\leq d}$ .

2.1F1F2step 1.1∎

If $i > j + d$ , then $i - j > d$ . The shift convention gives $H^{i} (F_{j, F} (X)) = H^{i - j} (W_{j, F} (X)) = 0$ by step 1.1. This is the claimed bound on the actual fiber. No conclusion has been extracted from a spectral sequence abutting to total K-groups.

Depends on

Used by · 2

Dependency tree · 3 levels

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