TheoremStatement: Literature-sourcedProof: Literature-sourced

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.

External input for simplicial Bloch cycle groups

PDF p. 4: Smooth comparison with Bloch cycles

Statement

Let $k$ be a field and $X$ a separated equidimensional smooth scheme of finite type over $k$ . For $r \geq 0$ , the free groups on codimension- $r$ integral closed subschemes of $X \times Δ^{n}$ meeting every coordinate face properly form a simplicial abelian group under face-intersection pullbacks with the cited local Tor multiplicities. In particular, their face maps obey $d_{i}^{n - 1} d_{j}^{n} = d_{j - 1}^{n - 1} d_{i}^{n}$ for $i < j$ , as equalities of cycles before quotienting by rational equivalence.

External input

This is the smooth separated equidimensional restriction of Definition 17.1 and the intersection convention in Appendix 17A. The compatibility of successive intersection multiplicities is imported; no local reconstruction of general intersection theory is claimed.

Depends on

Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture Notes on Motivic Cohomology · Definition 17.1, printed p. 135, PDF p. 147; Definition 17A.1 and Example 17A.4, printed pp. 143–144, PDF pp. 155–156; convention on printed p. 3, PDF p. 15.

Used by · 1

Admissible face restrictions and their cycle identities

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture Notes on Motivic Cohomology · Definition 17.1, printed p. 135, PDF p. 147; Definition 17A.1 and Example 17A.4, printed pp. 143–144, PDF pp. 155–156; convention on printed p. 3, PDF p. 15.

For separated equidimensional smooth finite-type schemes over a field, the groups generated by codimension-r integral cycles meeting simplex faces properly form a simplicial abelian group under intersection pullbacks, with the local Tor intersection multiplicities of Appendix 17A.

Hypotheses: A field k, a separated equidimensional smooth finite-type k-scheme X, nonnegative r and simplex degree, and proper intersection with all faces.

Reading limits: Inspected Definition 17.1 and its surrounding passage, the displayed Appendix 17A.1 and 17A.4 passages, and the separatedness convention in the accessible PDF extraction. The construction and intersection theory are imported; their proofs and the cited Bloch, Fulton and Serre sources were not reconstructed or independently read.

Sources

Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture Notes on Motivic Cohomology · Definition 17.1, printed p. 135, PDF p. 147; Definition 17A.1 and Example 17A.4, printed pp. 143–144, PDF pp. 155–156; convention on printed p. 3, PDF p. 15.