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Weight zero is the locally constant rank function

PDF p. 9: Actual low-weight motivic projections and their normalization

Statement

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied constructions. Let $k$ be any field, and $X$ a smooth finite-type $k$ -scheme. Under the canonical weight-zero comparison, the actual EM map $π_{0} q_{0} : K_{0} (X) \to H^{0} (X, Z (0)_{m o t})$ is the locally constant rank function, component by component. Consequently $π_{0} {Fil}^{1} K (X) = \ker (rank : K_{0} (X) \to H^{0} (X, Z))$ , and $π_{1} {Fil}^{1} K (X) \to K_{1} (X)$ is an isomorphism. These are statements about stage groups, not an assumed equality of a stage with its image.

Facts & Assumptions

[A1]

The assumed The Axiom of Choice and fixed bounds discharge the selected low-weight tower in [F1] and the affine perfect- $K$ comparison and filtration in [F2], used in steps 1.1–2.1. The pointwise unital rank test and the long exact sequence introduce no further choice.

Given: The Axiom of Choice, the fixed compatible bounds, smooth finite-type $X / k$ , all fields in their prime-field category, and the actual unital multiplicative EM tower.

Goal: Normalize $q_{0}$ and identify its first-stage homotopy groups.

[F1]

External EM low-weight maps with fixed positive normalization supplies the canonical ${gr}^{0}$ comparison and the unital $q_{0}$ ; its target on a smooth scheme is the ordinary locally constant integer sheaf in degree zero. Negative cohomology of this sheaf vanishes.

[F2]

Affine perfect K zero and bundle K zero identifies degree-zero perfect K-theory with finite-projective classes on affines, so $K_{0}$ of a field is $Z$ with generator its unit line. Motivic filtration and image groups supplies the actual exact fiber sequence ${Fil}^{1} K \to K \to {gr}^{0} K$ .

Proof

1.1A1F1F2

On a field extension $L$ of the prime field, $K_{0} (L) = Z$ generated by $[L]$ . The graded-zero comparison is unital, so the unital homomorphism $π_{0} q_{0} : Z \to Z$ is identity. For $x \in X$ , naturality of $q_{0}$ along $Spec κ (x) \to X$ therefore evaluates $q_{0} (α)$ as the rank of $α$ at $x$ . On a smooth $X$ , the weight-zero target is the locally constant integer function sheaf, so its sections are determined by these values at points. Thus $q_{0}$ equals the full rank function, including on disconnected $X$ .

2.1A1F1F2step 1.1∎

In the fiber sequence of [F2], $π_{i} {gr}^{0} K = H^{- i} (X, Z) = 0$ for $i > 0$ by [F1]. Its long exact sequence therefore identifies $π_{1} {Fil}^{1} K$ with $K_{1}$ , and identifies $π_{0} {Fil}^{1} K$ injectively with the kernel of $π_{0} q_{0}$ . Apply step 1.1 to obtain precisely the two asserted identifications.

Depends on

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