LemmaStatement: AI-generatedProof: AI-generated

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Agreement with the existing projective-relation and virtual bundle first classes

PDF p. 8: Proposition3.2, perfect-complex first class and determinant

Statement

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied motivic Chern, Picard and flag constructions. For any field $k$ and qcqs $X / k$ , the supplied projective-relation first class of an actual bundle $E$ satisfies $c_{1} (E) = γ_{X} (\det E)$ , where $γ_{X} : Pic (X) \to H^{2, 1} (X)$ is the actual EM orientation. The same holds for the existing virtual first class on $K_{v b}^{0} (X)$ . This statement neither assumes nor concludes that $K_{v b}^{0} (X) \to K_{0} (Perf (X))$ is surjective.

Facts & Assumptions

[A1]

Assume The Axiom of Choice and the stated site and universe bounds. These are inherited by the Whitney, projective-relation, flag-detection and virtual-series inputs in [F1]–[F3] when applied in steps 1.1–3.1. The finite rank partition introduces no additional arbitrary choice.

[F1]

Whitney identities and virtual Chern arithmetic supplies actual filtered-root Whitney and normalization on lines; The actual EM first Chern class is additive and natural supplies additive natural $γ$ .

[F2]

Determinants and exact sequences identifies the determinant of a line-filtered bundle with the tensor product of its line quotients, naturally under pullback.

[F3]

The abstract constant-rank detection statement in Detection on a simultaneous flag tower applies to the actual theory of Motivic projective bundle formula over every field, since its projective-bundle contract has that scope. Its finite-clopen motivic restriction property supplies varying ranks; Total Chern series for actual and virtual bundles defines their gluing and virtual extension.

Proof

Given: $E$ or a virtual bundle class on a qcqs scheme over an arbitrary field $k$ , with AC and the stated compatible size bounds.

Goal: Reuse the sound line/filtration argument with justified all-field detection.

1.1A1F1F2F3

For a constant-rank $E$ , take its flag tower in [F3]. Pullback is injective in bidegree(2,1). On the tower, [F1] makes $c_{1}$ the sum of its line-quotient first classes, and [F2] makes its determinant the tensor product of those lines. Additivity of $γ$ identifies these expressions. Naturality and injectivity therefore give $c_{1} (E) = γ_{X} (\det E)$ on $X$ . This uses the abstract detecting PBF statement, not an algebraically closed hypothesis of a separately worded splitting theorem.

2.1A1F1F2F3step 1.1

For varying rank use the finite clopen rank partition and joint restriction isomorphism in [F3]. The same equality holds on every component, including rank-zero components where both classes are zero. It therefore holds globally.

3.1A1F1F2F3step 2.1∎

For a difference $[V] - [W]$ in the presented vector-bundle group, the linear coefficient of its representative-independent series is $c_{1} (V) - c_{1} (W)$ by [F3]. The exact-sequence determinant identity in [F2] kills every relation of the presented bundle group, so determinant defines a homomorphism on that group without a locally Noetherian hypothesis. Together with additivity of $γ$ in [F1], this identifies the expression as $γ (\det V \otimes (\det W)^{- 1})$ . This proves the claimed virtual comparison without identifying either group with all perfect classes.

Depends on

Used by · 2

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