LemmaStatement: AI-generatedProof: AI-generated

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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.

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Combining the actual motivic and cdh outer-term bounds

PDF p. 4: Motivic dimension vanishing

Statement

For a prime field $F$ , a Noetherian $F$ -scheme $X$ of finite dimension $d$ and $j \geq 0$ , the complex $M_{j, F} (X)$ is supported in degrees $\leq j + d$ . Moreover the actual comparison $M_{j, F} (X) \to C_{j, F} (X)$ induces a surjection in degree $j + d$ and isomorphisms in every larger degree.

Facts & Assumptions

Given: $F, X, d, j$ under the stated hypotheses.

Goal: Combine the two specific outer terms, rather than replace graded vanishing by a claim about total K-theory.

[F1]

Prime-field evaluation and compatibility with the standing theory supplies the actual fiber sequence $F_{j, F} (X) \to M_{j, F} (X) \to C_{j, F} (X)$ .

[F2]

The global dimension bound for the specified comparison fiber gives the first outer bound $\leq j + d$ . External prime-field cdh comparison and cdh dimension bound supplies the second outer bound on exactly the Noetherian finite-dimensional prime-field scheme $X$ .

[F3]

Gluing upper bounds through an actual fiber sequence is the derived long-exact-sequence calculation.

Proof

1.1F1F2F3

Apply [F3] to the actual sequence in [F1] with $N = j + d$ , using the fiber bound from [F2]. It gives the surjection at degree $j + d$ and comparison isomorphisms for every $i > j + d$ . These are maps induced by the actual natural transformation, not merely abstract group isomorphisms.

2.1F1F2F3step 1.1∎

The cdh outer bound in [F2] makes the target of each high-degree comparison zero, so step 1.1 gives $H^{i} (M_{j, F} (X)) = 0$ for $i > j + d$ . Both source-side dimension inputs were applied with their full Noetherian and finite-dimensional hypotheses; $j = 0$ is included.

Depends on

Used by · 1

Motivic dimension vanishing

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