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.

Degree-zero Bloch cycles are ordinary cycles

PDF p. 13: CH3 identification and degree seven vanishing

Statement

Let $k$ be a field, $X$ a smooth finite-type $k$ -scheme, and $r \geq 0$ . Write $Z^{r} (X)$ for the free abelian group on integral closed subschemes of codimension $r$ in $X$ , with codimension measured on the ambient irreducible component containing the generic point. Then the identification $X \times Δ^{0} = X$ gives $z^{r} (X, 0) = Z^{r} (X)$ .

Statement references

Bloch cycle complexes and higher Chow groups

Facts & Assumptions

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

Goal: Identify the two free abelian groups on their generators.

[F1]

The definition of the Bloch complex uses integral closed codimension- $r$ subschemes meeting all nonempty coordinate faces in codimension $r$ . Bloch cycle complexes and higher Chow groups

Proof

1.1F1

The equation defining $Δ^{0}$ is $t_{0} = 1$ , so $X \times Δ^{0} = X$ . Its only nonempty coordinate face is itself, and every codimension- $r$ generator meets this face in codimension $r$ .

2.1step 1.1∎

Thus the two displayed free abelian groups have exactly the same generators, and the identity on those generators is the asserted identification.

Depends on

Bloch cycle complexes and higher Chow groups

Used by · 3

Dependency tree · 3 levels

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