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.

Open restriction of admissible cycles

PDF p. 13: Supported Chow group

Statement

Let $k$ be a field, $U$ a smooth finite-type $k$ -scheme, $V \subset U$ open and $r \geq 0$ . Restricting each admissible generator to $V \times Δ^{n}$ , with empty restriction sent to zero, defines a chain map $j^{*} : z^{r} (U, ∙) \to z^{r} (V, ∙)$ .

Statement references

Bloch cycle complexes and higher Chow groups

Facts & Assumptions

Given: $k, U, V, r$ as stated.

Goal: Establish admissibility and compatibility with boundaries.

[F1]

The cycle groups, face maps and their coefficients are defined by admissible intersections and local lengths. Bloch cycle complexes and higher Chow groups

Proof

1.1givenF1

A nonempty open subscheme of an integral scheme is integral and dense. Restricting a generator and its intersections with coordinate faces to the open subset $V$ preserves their codimensions at their generic points; its restriction is therefore an admissible generator, or zero if empty.

2.1F1step 1.1∎

Formation of a face intersection commutes with restriction to the open subset. At each surviving generic point the local rings and the length in the face coefficient are unchanged; components disjoint from the open disappear. Thus each face map commutes with restriction, and so does their alternating sum.

Depends on

Bloch cycle complexes and higher Chow groups

Used by · 3

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.