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Stable triviality forces vanishing of the actual Chern coefficients

Statement

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied motivic Chern constructions. If $E$ is stably trivial on a qcqs scheme over a field, all its positive motivic Chern classes vanish. If the scheme is smooth and equidimensional of finite type, all its positive ordinary integral Chow Chern classes vanish too. In either theory, this follows by applying the actual Whitney formula to a stable trivialization, without assuming that Chern classes of actual bundles depend only on an unspecified K-zero class.

Facts & Assumptions

[A1]

Assume The Axiom of Choice and the stated compatible site and universe bounds. These are inherited from the actual motivic Chern and Whitney suppliers when they are applied below; their finite direct-sum and coefficient calculations add no further choice principle.

Given: The data and hypotheses in the statement, AC and the stated compatible site and universe bounds for the actual motivic constructions.

Goal: Establish the stated conclusion.

[F1]

There is an actual stable isomorphism $E \oplus O_{X}^{N} ≅ O_{X}^{r + N}$ . Stable triviality and stable bundle equivalence

[F2]

In each of the stated theories trivial bundles have the unit Chern series. The actual Chern series of a trivial bundle is the unit

[F3]

The actual motivic and Chow direct-sum Whitney formulas hold, and Chern classes are invariant under bundle isomorphism. Whitney identities and virtual Chern arithmetic, Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes, Operational Chern classes and the Whitney formula, Chern classes of a vector bundle on a smooth scheme

Proof

1.1A1F1F2F3

Apply the actual direct-sum formula to the isomorphism in [F1]. Isomorphism invariance and [F2] give $c_{t} (E) \cdot 1 = c_{t} (E) c_{t} (O_{X}^{N}) = c_{t} (O_{X}^{r + N}) = 1$ . This argument applies independently to both the motivic and the ordinary Chow operations under their stated hypotheses.

2.1A1step 1.1algebra∎

Equality of formal series is equality of coefficients. Hence $c_{i} (E) = 0$ for every $i \geq 1$ in each applicable coefficient group. No cancellation of bundles and no comparison inferred from equality of groups has entered this deduction.

Depends on

Used by · 1

A nontrivial rank-two bundle with trivial Chern classes on a smooth complex affine fourfold

Dependency tree · 3 levels

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