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Second projection is minus second Chern class

PDF p. 11: Proposition 5.5; pp. 11–12

Statement

Assume AC and the fixed site/universe bounds. Let $k$ be algebraically closed and $X$ a normal qcqs $k$ -scheme. For perfect-complex $K$ -theory, the actual second-stage projection transports uniquely to its image subgroup, and $p_{2} : F_{m o t}^{2} K_{0} (X) ⟶ H_{m o t}^{4, 2} (X, Z), p_{2} (α) = - c_{2} (α) .$ Its kernel is $F_{m o t}^{3} K_{0} (X)$ . No finite-type or global bundle-representation hypothesis is imposed on this equality or kernel assertion. If, additionally, $X$ is finite type and $\dim X \leq 3$ , it induces $F^{2} / F^{3} ≅ H_{m o t}^{4, 2} (X, Z)$ with this sign.

Facts & Assumptions

Given: The normal qcqs scheme and the actual motivic stages and perfect operations.

Goal: Preserve the original scope, actual map, sign, kernel, and the existing finite-type low-dimensional supplement.

[F1]

Actual second-stage comparison on every qcqs field-scheme identifies the actual second-layer map with minus the actual second Chern map as derived sheaf maps on every qcqs field-scheme. Its determinant-fiber retraction also proves injectivity of the second-stage augmentation on $π_{0}$ , without finite-type assumptions.

[F2]

Motivic filtration and image groups defines image groups and their compatible augmentations. Stage projections, kernels and image groups supplies exactness of the consecutive stage sequence. Second Chern additivity on the second piece gives additivity in this full scope.

[F3]

The low-dimensional quotient follows from the stage sequence gives the quotient isomorphism for normal finite-type schemes of dimension at most three.

[F4]

The Axiom of Choice states the Choice assumption inherited from the actual filtration, section and Chern-operation constructions in [F1]–[F3].

Proof

1.1F1F2F4

By [F1] the second-stage augmentation is injective on $π_{0}$ , hence identifies that group with its image $F^{2}$ . Transport the actual layer map by this isomorphism to define $p_{2}$ . Evaluating the actual map identity of [F1] on the unique lift of any $α \in F^{2}$ gives $p_{2} (α) = - c_{2} (α)$ ; it includes non-bundle-representable perfect classes. Additivity is also consistent with [F2].

2.1F2step 1.1

Exactness of the third-to-second stage sequence in [F2] says that the layer value of the lift is zero precisely when it comes from a third-stage class. Augmenting to $K_{0}$ identifies these classes exactly with the third image subgroup. Conversely any third-stage representative has zero second-layer value. Injectivity at stage two from step 1.1 therefore gives $\ker (p_{2}) = F^{3}$ ; no injectivity at stage three is assumed.

3.1F3step 1.1step 2.1∎

Under the additional finite-type and dimension assumptions apply [F3]. It gives surjectivity of this same actual projection, and steps 1.1–2.1 identify its sign and kernel. Thus the quotient supplement is retained without narrowing the normal qcqs equality.

Depends on

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