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High filtration vanishing

PDF p. 9: Lemma 4.2

Statement

Assume the Axiom of Choice. Let $X$ be a Noetherian scheme of finite Krull dimension $d$ over a field $k$ , and let $i$ and $j$ be integers. For the motivic filtration of Motivic filtration and image groups, $π_{j} F_{m o t}^{i} K (X) = 0 if i \geq d and j < 0,$ and $π_{0} F_{m o t}^{i} K (X) = 0 if i > d .$ Here $F_{m o t}^{i} K (X)$ is the $i$ th spectrum in the decreasing motivic tower of nonconnective $K$ -theory, and $π_{j}$ is its $j$ th stable homotopy group. These are assertions about the stage groups themselves, not merely their images in $K_{j} (X)$ . The field is arbitrary; in applying the fixed-field definitions below, take the standing field to be $k$ .

Facts & Assumptions

Given: The Axiom of Choice, a Noetherian finite-dimensional $k$ -scheme $X$ , and integers in either of the two stated ranges.

Goal: Prove the two stage-group vanishing assertions.

[F1]

Motivic filtration and image groups supplies the actual decreasing tower on qcqs schemes over an arbitrary specified field, with the stated choice assumption. A Noetherian scheme is quasi-compact and quasi-separated.

[F2]

For $r \geq 0$ , Motivic graded pieces and spectral sequence identifies the actual cofiber of $F_{m o t}^{r + 1} K (X) \to F_{m o t}^{r} K (X)$ with $Z (r)_{m o t} (X) [2 r]$ . By Complex shifts and spectral homotopy indices, its $j$ th homotopy group is $H_{m o t}^{2 r - j, r} (X, Z)$ .

[F3]

Motivic dimension vanishing gives $H_{m o t}^{a, r} (X, Z) = 0$ for $r \geq 0$ and $a > r + d$ . Its equicharacteristic hypothesis holds because $X$ is a scheme over a field, hence over that field's prime subfield.

[F4]

Stage projections, kernels and image groups gives exactness of $π_{j} F^{r + 1} \to π_{j} F^{r} \to π_{j} {gr}^{r}$ for a decreasing filtered spectrum. Eventual homotopy vanishing of the motivic tower gives, for each fixed $j$ , vanishing of $π_{j} F_{m o t}^{N} K (X)$ at all sufficiently large $N$ ; it also records vanishing of the stages on the empty scheme.

Proof

1.1F1F3F4given

If $X$ is empty, the stage spectra vanish by [F4], and both assertions follow. Otherwise $d \geq 0$ . Since a Noetherian scheme is qcqs and its structure map factors through the prime field of $k$ , all hypotheses of [F1]–[F4] apply.

2.1F2F3F4givenstep 1.1algebra

Fix $j < 0$ and $i \geq d$ . For every integer $r \geq i$ , we have $r \geq 0$ and $2 r - j > r + d$ , because $r - d \geq 0$ and $- j > 0$ . By [F2] and [F3], $π_{j} {gr}_{m o t}^{r} K (X) = 0$ . Exactness in [F4] therefore makes each transition $π_{j} F_{m o t}^{r + 1} K (X) \to π_{j} F_{m o t}^{r} K (X)$ surjective.

3.1F4step 2.1

Choose $N \geq i$ large enough that $π_{j} F_{m o t}^{N} K (X) = 0$ , using [F4]. The finite composite of the surjections from stage $N$ to stage $i$ is surjective, so its target is zero. This proves the negative-degree assertion.

4.1F2F3F4givenstep 1.1algebra∎

Now let $j = 0$ and $i > d$ . For every $r \geq i$ , we have $r \geq 0$ and $2 r > r + d$ . The same application of [F2]–[F4] makes every transition $π_{0} F_{m o t}^{r + 1} K (X) \to π_{0} F_{m o t}^{r} K (X)$ surjective. Eventual vanishing in degree zero supplies an $N \geq i$ with zero source; composing down to stage $i$ proves $π_{0} F_{m o t}^{i} K (X) = 0$ . Thus both stated assertions hold.

Depends on

Used by · 6

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 9, Lemma 4.2 and proof

For a Noetherian scheme of dimension d over a field, the indicated negative homotopy groups vanish at stages i at least d, and degree zero vanishes at stages i greater than d.

Hypotheses: A Noetherian scheme of finite dimension d over a field.

Reading limits: The lemma statement and its proof were read in the supplied article text. The present proof uses the completed local supplier contracts for the graded comparison, dimension bound and eventual stage vanishing.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 9, Lemma 4.2 and proof