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Motivic dimension vanishing

PDF p. 4: Motivic dimension vanishing

Statement

Let $X$ be a Noetherian equicharacteristic scheme of finite dimension $d$ , and let $j \geq 0$ . Then $H_{m o t}^{i, j} (X, Z) = 0 for every i > j + d .$ Here equicharacteristic has the precise EM meaning: the structure map $X \to Spec Z$ factors through $Spec F$ for a prime field $F = Q$ or $F_{p}$ . For such $X$ the groups mean $H^{i} (M_{j, F} (X))$ in the general prime-field interface. If $X$ is over the standing field $k$ , this interface is exactly the theory already defined on ${S c h}_{k}$ , by restriction along its prime subfield. The theorem does not require that every $X$ be over that one standing field.

Statement references

Motivic complexes and cohomology

Prime-field evaluation and compatibility with the standing theory

Combining the actual motivic and cdh outer-term bounds

Facts & Assumptions

Given: The stated Noetherian finite-dimensional globally equicharacteristic scheme, nonnegative weight and the compatible actual EM interface.

Goal: Prove the unrestricted prime-field statement and identify its standing fixed-field specialization.

[F1]

Prime-field evaluation and compatibility with the standing theory defines the actual general EM groups and proves compatibility with the completed fixed-field definition.

[F2]

Normalizing the comparison fiber for the dimension criterion and The global dimension bound for the specified comparison fiber check and apply the imported geometric dimension criterion to the actual shifted comparison fiber.

[F3]

Gluing upper bounds through an actual fiber sequence and Combining the actual motivic and cdh outer-term bounds combine the actual fiber bound with the precise cdh outer bound through the long exact sequence.

Proof

1.1F1F2F3

If $X$ is nonempty, its assumed equicharacteristic factorization makes it a scheme over the corresponding prime field $F$ . The Noetherian and finite-dimensional assumptions are exactly those of [F2]–[F3]. Apply [F3] to obtain $H^{i} (M_{j, F} (X)) = 0$ whenever $i > j + d$ . By [F1] these are the claimed general motivic groups. For the empty scheme the complexes vanish by empty-cover descent, so the assertion is immediate.

2.1F1F3step 1.1∎

When $X$ additionally carries the standing $k$ -structure, [F1] identifies this very complex and its pullbacks with those of the standing definition by actual restriction, so step 1.1 gives its stated specialization. No finite-type, smoothness, regularity or fixed- $k$ restriction has replaced the original Noetherian equicharacteristic hypotheses. This proves the theorem for exactly the defined range $j \geq 0$ .

Depends on

Used by · 1

High filtration vanishing

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

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.13 p.8 and Theorem8.1 p.53

For j≥0 and finite-dimensional Noetherian equicharacteristic X, motivic cohomology vanishes in degrees i>j+dimX.

Hypotheses: Global equicharacteristic factorization through SpecQ or SpecF_p; X Noetherian and finite-dimensional; j≥0.

Reading limits: Read these full statements; the present numbered argument uses the dedicated imported cdh/fiber bounds and dimension criterion, with their full reading limits, rather than reconstructing every geometric foundation.

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p.4 property(3), referring to EM23 Theorem1.13

The article invokes the full motivic dimension vanishing bound.

Hypotheses: The precise theorem hypotheses are retained from the referenced EM theorem.

Reading limits: The target attribution and original claim were retained from the native plan; the exact supporting EM passages were read directly for this authoring task.

Sources

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.13 p.8 and Theorem8.1 p.53
Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p.4 property(3), referring to EM23 Theorem1.13