LemmaStatement: AI-generatedProof: AI-generated

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Third Chern additivity on the third piece

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

Statement

Assume AC and the fixed site/universe bounds. Let $k$ be the standing algebraically closed field and let $X$ be any normal qcqs $k$ -scheme. For the specified perfect-complex K-theory and actual motivic Chern operations, $c_{3} : F_{m o t}^{3} K_{0} (X) ⟶ H_{m o t}^{6, 3} (X, Z)$ is a homomorphism. No finite-type, noetherian, finite-dimensional, reduced-presentation or global bundle-resolution hypothesis is added. In fact the proof supplies the same assertion on every qcqs scheme over any field.

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 lower-class vanishing in [F2] at step 1.1 and the actual coefficient calculation in [F3] at step 2.1.

[F1]

Article scheme conventions fixes the original normal qcqs scope; Motivic filtration and image groups defines the third image subgroup in this scope.

[F2]

The first two perfect Chern classes vanish on the third image piece in full scope proves $c_{1} = c_{2} = 0$ for every class in that subgroup using actual stage maps in full scope.

[F3]

Degree-three coefficient algebra on the lower-Chern kernel proves the actual degree-three coefficient calculation after the full-scope Cartan construction.

Proof

Given: The normal qcqs scheme in the statement and $x, y \in F_{m o t}^{3} K_{0} (X)$ .

Goal: Preserve the article's full scope and show homomorphy for actual perfect classes.

1.1A1F1F2

Apply [F2] to $x, y$ . Their first and second perfect Chern classes vanish, with no image-stage injectivity or finite-type hypothesis.

2.1A1F3step 1.1∎

Apply [F3] to this subgroup. The degree-three Cartan cross terms vanish by step 1.1, yielding $c_{3} (x + y) = c_{3} (x) + c_{3} (y)$ and preservation of zero and negatives. Thus the displayed map is a homomorphism in the exact original scope.

Depends on

Used by · 1

Third Chern isomorphism

Dependency tree · 2 levels

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