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.

Negative homology of the cycle complex

PDF p. 13: CH3 identification and degree seven vanishing

Statement

For a field $k$ , a smooth finite-type $k$ -scheme $X$ , $r \geq 0$ , and an integer $n < 0$ , one has $H_{n} z^{r} (X, ∙) = 0$ .

Facts & Assumptions

Given: $k$ , $X$ , $r$ and $n$ as in the statement.

Goal: Compute the indicated homology group.

[F1]

The Bloch complex has zero chain group in every negative degree, and its homology is the kernel modulo the image. Bloch cycle complexes and higher Chow groups

Proof

1.1F1

Since $n < 0$ , the chain group in degree $n$ is zero, so the kernel of its outgoing differential is zero.

2.1F1step 1.1∎

The incoming image is a subgroup of this zero kernel; consequently their quotient is zero.

Depends on

Bloch cycle complexes and higher Chow groups

Used by · 2

Dependency tree · 3 levels

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