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.

Reindexing a comparison with higher Chow groups

PDF p. 13: CH3 identification and degree seven vanishing

Statement

Let $k$ be a field, $X$ a smooth finite-type $k$ -scheme and $r \geq 0$ . Suppose that for every integer $p$ a group $H^{p, r} (X)$ is equipped with an isomorphism $H^{p, r} (X) ≅ {C H}^{r} (X, 2 r - p)$ . Then $H^{2 r, r} (X) ≅ H_{0} z^{r} (X, ∙)$ and $H^{p, r} (X) = 0$ for $p > 2 r$ . No existence of such comparison isomorphisms is asserted.

Facts & Assumptions

Given: $k$ , $X$ , $r$ and the stated isomorphisms.

Goal: Identify the index-zero case and the negative-index range.

[F1]

Higher Chow groups are the homology groups of the Bloch complex. Bloch cycle complexes and higher Chow groups

[F2]

The homology of that complex vanishes in negative degrees. Negative homology of the cycle complex

Proof

1.1givenF1

Substituting $p = 2 r$ makes the homological index zero, giving the first isomorphism by the definition of higher Chow groups.

2.1givenF1F2∎

If $p > 2 r$ , then $2 r - p < 0$ ; the target of the assumed isomorphism is therefore zero, and so is its source.

Depends on

Used by · 1

Degree zero and negative homology of the cycle complex

Dependency tree · 3 levels

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