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 full top-weight map with its exact factorization

PDF p. 6: Proposition 2.6 proof

Statement

For $X$ finite type over algebraically closed $k$ , $\dim X \leq 3$ and $p$ invertible, let $j = 3, n = p$ . The canonical map $β_{X} : Z (3)_{m o t} (X) / p \to B_{3} (X)$ is an equivalence and factors as $β_{X} = ϵ_{X} γ_{X}$ , where $ϵ$ is the source map induced from $τ^{\leq 3} B_{3} \to B_{3}$ by cdh localization. This statement uses the actual top-weight source comparison, rather than an arbitrary equivalence of complexes.

Facts & Assumptions

Given: The hypotheses in the statement.

Goal: Establish the stated exact map or descent conclusion.

[F1]

External canonical top-weight motivic-to-syntomic comparison.

[F2]

The actual composite and its objectwise cofiber and An independent full equivalence into the truncated cdh localization.

Proof

1.1F1

The draft top-weight supplier [F1], importing the published Elmanto report Theorem 4, applies to every finite-type scheme over algebraically closed $k$ , with $j \geq \dim X$ and any prime; its invertible-prime target is exactly $B_{3} (X)$ . With $j = 3$ its actual comparison $β_{X}$ is therefore an equivalence, without requiring affineness, smoothness or reducedness.

2.1F1F2step 1.1∎

The draft source-composite supplier [F2] identifies this canonical motivic-to-étale map as $ϵ θ α$ . This is the same prime-field comparison used in the published report Theorem3(1) and Theorem4, by their explicit assertion that the equivalence is under the motivic-to-syntomic comparison. The preceding lemma defines $γ = θ α$ , yielding the equation with actual maps. No equality is inferred from abstractly isomorphic groups.

Depends on

Used by · 2

Dependency tree · 3 levels

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