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.

Boundaries in degree zero

PDF p. 13: CH3 identification and degree seven vanishing

Statement

Under the hypotheses of Degree-zero Bloch cycles are ordinary cycles, the subgroup $B^{r} (X) = im (\partial_{1}) \subset Z^{r} (X)$ is generated by the differences $d_{0}^{1} [W] - d_{1}^{1} [W]$ as $W$ ranges over the admissible integral codimension- $r$ closed subschemes of $X \times Δ^{1}$ .

Facts & Assumptions

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

Goal: Describe the image of the degree-one boundary.

[F1]

The degree-one chain group is freely generated by the stated admissible subschemes, and $\partial_{1} = d_{0}^{1} - d_{1}^{1}$ . Bloch cycle complexes and higher Chow groups

[F2]

The degree-zero chain group is $Z^{r} (X)$ . Degree-zero Bloch cycles are ordinary cycles

Proof

1.1F1

By linearity, the image of a free abelian group under a homomorphism is generated by the images of its free generators.

2.1F1F2step 1.1∎

Applying the displayed formula for $\partial_{1}$ and identifying its target with $Z^{r} (X)$ gives precisely the stated generators of $B^{r} (X)$ .

Depends on

Used by · 2

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.