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Top-weight degree-five vanishing away from the characteristic

PDF p. 5: Proposition 2.6: torsion freeness

Statement

Let $X$ be an affine finite-type scheme of dimension three over an algebraically closed field $k$ . If $ℓ$ is invertible in $k$ , then $H_{m o t}^{5, 3} (X, Z / ℓ) = 0$ .

Facts & Assumptions

[F1]

External canonical top-weight motivic-to-syntomic comparison identifies the actual finite-coefficient motivic complex with syntomic cohomology in weight three, since $3 = \dim X$ ; at an invertible prime this is $R Γ_{\overset{ˊ}{e} t} (X, μ_{ℓ}^{\otimes 3})$ .

[F2]

External affine étale vanishing at invertible primes gives affine étale vanishing above dimension.

Proof

Given: $X, k, ℓ$ as stated.

Goal: Prove the vanishing for the supplied motivic theory.

1.1F1

The quotient in [F1] is the objectwise derived quotient used in Motivic complexes and cohomology, not a different finite-coefficient theory. Thus its degree-five cohomology is $H_{\overset{ˊ}{e} t}^{5} (X, μ_{ℓ}^{\otimes 3})$ .

2.1F2step 1.1∎

The twist is an étale $ℓ$ -torsion local system; [F2] makes this group zero since $5 > 3$ . This does not require $X$ reduced or smooth.

Depends on

Used by · 1

Torsion freeness of top threefold motivic cohomology

Dependency tree · 3 levels

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