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Unique divisibility of the smooth-point subgroup

PDF p. 17: Proof of Corollary 6.9, Kri19 Corollary 1.5

Statement

Assume the Axiom of Choice and the size conventions of the smooth-point construction. Let $k$ be an algebraically closed field and $X$ a normal, pure-dimensional affine $k$ -scheme of finite type and dimension $3$ . The smooth-point subgroup $F_{s m}^{3} K_{0} (X)$ , generated by the perfect residue-sheaf classes of smooth closed points, is uniquely divisible: multiplication by every positive integer is an isomorphism. In particular multiplication by $2$ is an isomorphism.

More generally, if normal affine $X$ has dimension $3$ but has lower-dimensional components, let $X_{3}$ be the union of its three-dimensional components. Its zero-extended subgroup $F_{s m, t o p}^{3} K_{0} (X) = (i_{3})_{*} F_{s m}^{3} K_{0} (X_{3})$ is uniquely divisible. This assertion does not include smooth points on lower-dimensional components in fixed weight $3$ .

Facts & Assumptions

Given: The field and scheme hypotheses, choice and size conventions in the statement.

Goal: Prove unique divisibility in the pure case and for the stated zero-extended subgroup, including the multiplier $2$ .

[F1]

Krishna's affine zero-cycle divisibility and kernel input gives unique divisibility of the Levine–Weibel group and a kernel killed by $2$ for its actual cycle map on a reduced pure affine threefold over $k$ .

[F2]

Levine–Weibel cycle image with pure-dimensional and component conventions identifies the actual cycle-map image with the smooth-point subgroup, using the residue-sheaf point classes. Subgroup generated by smooth closed points defines the latter for pure schemes.

[F3]

Divisibility of an image with torsion kernel gives unique divisibility of an image when its domain is uniquely divisible and its kernel is torsion.

[F4]

Dimension-specific point images and a precise lower-component obstruction supplies the pure top-dimensional stratum and the zero-extended subgroup; Finite clopen restrictions split the actual motivic image filtration makes zero extension an injective coordinate inclusion. Unique divisibility is invariant under group isomorphism transports unique divisibility across an isomorphism.

Proof

1.1F1F2F3given

A normal finite-type affine scheme is reduced, so the pure $X$ satisfies [F1]. The kernel killed by $2$ is torsion. Applying [F3] to the cycle map yields unique divisibility of its image. By [F2] that image is exactly $F_{s m}^{3} K_{0} (X)$ with the stated point-class normalization. Thus multiplication by every positive integer, in particular $2$ , is an isomorphism on that subgroup.

2.1F4step 1.1∎

In the dimension-only case, [F4] makes $X_{3}$ a pure normal affine threefold and identifies zero extension with an injective coordinate inclusion. Step 1.1 applies to $X_{3}$ ; its zero-extension map is an isomorphism onto the subgroup defined in the statement. The transport result in [F4] therefore gives unique divisibility there. The definition selects top-dimensional components, so it makes no fixed-weight assertion about points on smaller components.

Depends on

Used by · 2

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Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 17, proof of Corollary 6.9; p. 14, Definition 6.5
Amalendu Krishna, Murthy’s conjecture on 0-cycles · v3, Theorems 1.3 and 1.4, Corollary 1.5 and §2.1