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-three coefficient algebra on the lower-Chern kernel

PDF p. 14: Lemma6.4; scheme conventions p3

Statement

Assume the Axiom of Choice and the fixed site and universe bounds of the supplied constructions. On every qcqs field-scheme $X$ , the actual third Chern operation is additive on any subgroup of perfect $K_{0} (X)$ where $c_{1} = c_{2} = 0$ . In particular it is additive on $F_{m o t}^{3} K_{0} (X)$ ; it sends zero to zero and $- x$ to $- c_{3} (x)$ there.

Facts & Assumptions

[A1]

Assume the Axiom of Choice as formulated in The Axiom of Choice and the stated size bounds. Choice and the fixed size bounds are inherited from perfect Cartan in [F1] at steps 1.1–2.1 and lower-class vanishing in [F2] at step 2.1. The subsequent coefficient arithmetic is finite algebra.

[F1]

Actual perfect Cartan through degree three on all qcqs field-schemes supplies the coefficientwise identity through degree three for every perfect class in this scope, with even first degrees and the actual motivic cup product.

[F2]

The first two perfect Chern classes vanish on the third image piece in full scope gives $c_{1} = c_{2} = 0$ on $F^{3}$ . Formal Chern arithmetic through degree three records the formal inverse coefficient identity; its algebra also follows directly from [F1].

Proof

Given: A subgroup $G \subseteq K_{0} (X)$ with the stated vanishing, and $x, y \in G$ .

Goal: Derive the homomorphism assertion purely after the actual full-scope operation identities are available.

1.1A1F1given

Extract the third coefficient in [F1]: $c_{3} (x + y) = c_{3} (x) + c_{2} (x) c_{1} (y) + c_{1} (x) c_{2} (y) + c_{3} (y) .$ The two middle terms vanish by the hypothesis on $G$ . Hence $c_{3} (x + y) = c_{3} (x) + c_{3} (y)$ .

2.1A1F1F2step 1.1∎

The pointed operation gives $c_{3} (0) = 0$ . Applying step 1.1 to $x + (- x) = 0$ gives $c_{3} (- x) = - c_{3} (x)$ . Equivalently the formal inverse has third coefficient $- c_{3} + 2 c_{1} c_{2} - c_{1}^{3}$ , which reduces to $- c_{3}$ on $G$ . Apply [F2] with $G = F^{3}$ , a subgroup by the image definition.

Depends on

Used by · 1

Third Chern additivity on the third piece

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.