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Degreewise boundedness and finite collections of eventual vanishings

PDF p. 4: Motivic graded spectral sequence; PDF p. 9: Eventual filtration vanishing

Statement

Assume the Axiom of Choice. The actual motivic filtration of every finite-dimensional Noetherian equicharacteristic scheme is degreewise bounded above in filtration on all integer homotopy degrees. For any finite set $S \subseteq Z$ and any anchor $t$ , one may choose a common sufficiently large $ℓ$ making every $π_{q} F^{t + ℓ}$ with $q \in S$ zero. The same holds for their image subgroups in $K_{q} (X)$ . No threshold independent of an unbounded set of degrees is asserted.

Facts & Assumptions

Given: The Axiom of Choice and the general source filtration or its actual standing-field restriction.

Goal: Express the stagewise result in the completed local boundedness language and in the form used by finite exact-sequence arguments.

[F1]

For nonempty $X$ , Explicit eventual-stage thresholds in every fixed homotopy degree supplies an actual zero-group threshold for each fixed degree and anchor.

[F2]

Valuative dimension and degreewise bounded towers defines degreewise boundedness; Motivic filtration and image groups defines the image subgroups.

[F3]

The assumed The Axiom of Choice discharges the inherited Choice premises of the threshold supplier in [F1] and the selected filtration in [F2]. The finite maximum in the proof introduces no stronger choice assumption.

[F4]

Every actual motivic stage on the empty scheme is zero proves that every actual stage on the empty scheme is zero, independently of any dimension convention.

Proof

1.1F1F2F3F4given

If $X$ is empty, [F4] gives all stage and image vanishings, with threshold zero for any anchor. Suppose henceforth that $X$ is nonempty. Under the inherited Choice premise in [F3], take anchor $t = 0$ in [F1]. For every fixed $q$ , all stages beyond $ℓ_{0} (q, 0)$ have zero $π_{q}$ . These are exactly the quantifiers in [F2]. If a stage group is zero, its image under the augmentation to $K_{q} (X)$ is zero as well.

2.1F1F2step 1.1∎

For a finite nonempty $S$ , take the maximum of the finitely many $ℓ_{0} (q, t)$ ; the empty set is vacuous. Every larger $ℓ$ satisfies every individual threshold in [F1], proving simultaneous vanishing for that finite set. This construction leaves the dependence on $S$ and $q$ explicit and makes no assertion of a universal bound for all integer homotopy degrees.

Depends on

Used by · 1

Eventual homotopy vanishing of the motivic tower

Dependency tree · 3 levels

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