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The actual Chern series of a trivial bundle is the unit

Statement

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied motivic Chern constructions. For every $m \geq 0$ , the actual total motivic Chern series of the trivial bundle on a qcqs scheme over a field is $c_{t} (O_{X}^{m}) = 1$ . On a smooth equidimensional finite-type scheme the same identity holds for the ordinary integral Chow Chern series. These statements refer to the actual projective-relation classes and their Whitney formulas. A formal variable $t$ records degree, so $c_{t} (V) = 1 + \sum_{i \geq 1} c_{i} (V) t^{i}$ .

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]

The completed actual motivic Whitney formula and additive first-Chern orientation hold for these classes; a line has no higher classes and $c_{t} (0) = 1$ . Whitney identities and virtual Chern arithmetic, Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes, Motivic Chern classes

[F2]

The ordinary Chow classes have zero positive Chern classes for a trivial line, obey Whitney, and have $c_{t} (0) = 1$ . Chern classes of a vector bundle on a smooth scheme, Operational Chern classes and the Whitney formula

Proof

1.1A1F1F2algebra

For the motivic trivial line, tensor-additivity gives $c_{1} (O_{X}) = c_{1} (O_{X} \otimes O_{X}) = 2 c_{1} (O_{X})$ . Subtracting in its coefficient group yields zero. Higher line classes vanish, so its total series is 1. The ordinary Chow trivial line has the same unit series by [F2].

2.1A1F1F2step 1.1∎

For $m > 0$ , apply the respective actual direct-sum Whitney formula to the $m$ trivial line summands. The product is $1^{m} = 1$ . For $m = 0$ the zero-bundle convention gives the same result. Thus both assertions include all finite stabilizations.

Depends on

Used by · 2

Dependency tree · 3 levels

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