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Dimension-specific point images and a precise lower-component obstruction

PDF p. 3: Levine–Weibel image after TheoremC; PDF p. 14: Definition6.5 pure-dimensional smooth-point subgroup

Statement

Assume the Axiom of Choice and the fixed size conventions of the supplied constructions. Let $k$ be a perfect field. For a normal affine finite-type $k$ -scheme $X$ of dimension $d \geq 1$ , define the top-dimensional point image by $F_{s m, t o p}^{d} K_{0} (X) : = (i_{d})_{*} F_{s m}^{d} K_{0} (X_{d}),$ where $(i_{d})_{*}$ is the exact clopen zero-extension map justified above. This definition has no pure-dimensional hypothesis on $X$ ; its piece $X_{d}$ is pure. Its generators are precisely the residue classes of smooth closed points on the dimension- $d$ components. Then $im ({cyc}_{X}^{K r}) = F_{s m, t o p}^{d} K_{0} (X) \subseteq F_{m o t}^{d} K_{0} (X) .$ For each positive dimension $e$ , the analogous zero-extended image of ${CH}_{0}^{L W} (X_{e})$ is contained in $F_{m o t}^{e} K_{0} (X)$ . The subgroup generated by all geometric smooth closed points of $X$ is the direct sum of these dimension-specific point images, together with the free residue-point subgroup on $X_{0}$ in weight zero. It need not be contained in $F_{m o t}^{d} K_{0} (X)$ .

Facts & Assumptions

[A1]

The Axiom of Choice and the fixed size conventions of the cited constructions are inherited through Finite clopen restrictions split the actual motivic image filtration. They discharge that supplier’s local-realization premise at its first application below. The Axiom of Choice

Given: The stated normal affine finite-type scheme over a perfect field, the Axiom of Choice and the fixed size conventions of the cited constructions, together with the clopen dimension strata and their established cycle maps, all with AC.

Goal: Give the strongest componentwise statement and exclude the falsely broadened fixed-weight assertion.

[F1]

Levine–Weibel relations split on clopen pure-dimensional components identifies Krishna's source quotient with that on $X_{d}$ . The quotient cycle map has exactly the smooth-point image gives its pure image and generator normalization.

[F2]

Normal dimension strata and the actual residue class model and Finite clopen restrictions split the actual motivic image filtration give the actual clopen K-class/image-filtration decompositions.

[F3]

Weight zero is the locally constant rank function gives $F_{m o t}^{1} K_{0} (k) = 0$ since rank $K_{0} (k) = Z \to Z$ is identity; Motivic filtration and image groups gives nesting and pullback naturality.

Proof

1.1A1F1F2

Use the supplied realization under [A1]. Each nonempty $X_{e}$ is a normal affine finite-type scheme of pure dimension $e$ over the same perfect field, so the corrected pure quotient-image lemma applies whenever $e > 0$ . Under [F1], the source point map on $X$ has exactly the same generator values as the map on $X_{d}$ followed by clopen zero extension. Thus its image is the displayed top-dimensional point image by the quotient-image argument. [F2] puts it inside $F_{m o t}^{d} K_{0} (X)$ because it lies in the dimension- $d$ piece there and is zero in every other component. Repeating with $e$ gives the same-weight containments for every positive-dimensional stratum. The displayed definition is legitimate because [F2] supplies the exact zero-extension map and its injective coordinate inclusion.

1.2F2algebra

Smooth closed points partition by the finitely many strata, so their generated subgroup splits by [F2] as the direct sum of the component generated subgroups. A normal zero-dimensional affine finite-type scheme over the perfect field is a finite disjoint union of spectra of finite separable extensions; its residue classes generate its $K_{0}$ with weight-zero rank generators. This last point description uses the free point group, not Krishna's $d \geq 1$ definition on a zero-dimensional scheme. Thus no convention for a nonexistent positive-codimension curve is being imposed on $X_{0}$ .

2.1F2F3∎

For a concrete failure of the all-points fixed-weight extension, let $X = A_{k}^{3} ⨿ Spec k$ over algebraically closed $k$ . It is normal affine of dimension three. Its isolated component point $p$ is geometrically smooth, and its class restricts to $1 \in K_{0} (k) = Z$ . Every class of $F_{m o t}^{3} K_{0} (X)$ restricts to $F_{m o t}^{3} K_{0} (k) \subseteq F_{m o t}^{1} K_{0} (k) = 0$ by [F3]. Therefore $[p] \notin F_{m o t}^{3} K_{0} (X)$ . This proves the obstruction to using all lower-dimensional smooth points at weight three, without asserting a defect in an implicit pure-threefold source convention.

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