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Characteristic-zero coefficient vanishing and torsion freeness

PDF p. 6: Proof of Proposition 2.6, multiplication by p sequence and conclusion

Statement

Let $k$ be an algebraically closed field of characteristic zero and let $X$ be an affine threefold over $k$ . For every prime $p$ , choosing a primitive $p$ th root of unity in $k$ gives an isomorphism $H_{m o t}^{5, 3} (X, Z / p) ≅ H_{e t}^{5} (X, Z / p) = 0.$ Consequently multiplication by $p$ is injective on $H_{m o t}^{6, 3} (X, Z)$ for every prime $p$ , and this group is torsion-free.

Here affine threefold means $X = Spec A$ , with $A$ a finitely generated $k$ -algebra and $\dim X = 3$ , as in Article scheme conventions; no smoothness, reducedness, or irreducibility is assumed. The motivic groups are the cohomology of the Elmanto–Morrow weight-three complex, with finite coefficients formed by derived tensor product after evaluation, as defined in Motivic complexes and cohomology. The étale group is the cohomology of the small étale site of $X$ with the constant sheaf $Z / p$ . Torsion-free means that no nonzero element is annihilated by a positive integer.

Facts & Assumptions

Given: An algebraically closed characteristic-zero field $k$ , an affine finite-type $k$ -scheme $X$ of dimension three, and a prime $p$ .

Goal: Establish the displayed isomorphism and vanishing, the coefficient-sequence injectivity, and torsion-freeness.

[F1]

Motivic complexes and cohomology defines $C = Z (3)_{m o t} (X)$ , $H_{m o t}^{i, 3} (X, Z) = H^{i} (C)$ , and $H_{m o t}^{i, 3} (X, Z / p) = H^{i} (C \otimes^{L} Z / p)$ .

[F2]

External canonical top-weight motivic-to-syntomic comparison gives, for finite-type $X$ over an algebraically closed field and weight $j \geq \dim X$ , an equivalence from the actual finite-coefficient motivic complex to $R Γ_{e t} (X, μ_{p}^{\otimes j})$ when $p$ is invertible in the field. Here $μ_{p}$ is the étale sheaf of $p$ th roots of unity and $μ_{p}^{\otimes j}$ is its $j$ -fold tensor power over $Z / p$ .

[F3]

A chosen root trivializes the actual Tate coefficient on the slice says that a chosen primitive $p$ th root $ζ \in k$ induces an isomorphism of étale sheaves $Z / p \tilde{\to} μ_{p}^{\otimes 3}$ , sending $1$ to $ζ^{\otimes 3}$ . External affine étale vanishing at invertible primes gives $H_{e t}^{i} (X, μ_{p}^{\otimes 3}) = 0$ for $i > \dim X$ under the present affine, finite-type and invertibility hypotheses.

[F4]

Derived finite coefficients and the Bockstein sequence supplies the exact sequence $H^{5} (C / p) \to H^{6} (C) \overset{p}{\to} H^{6} (C)$ , where $C / p = C \otimes^{L} Z / p$ . Primewise detection of torsion says an abelian group is torsion-free if multiplication by every prime is injective.

Proof

1.1givenF3construct

Characteristic zero makes $p$ invertible in $k$ . Since $k$ is algebraically closed, $T^{p} - 1$ splits in $k$ ; its roots are distinct because its derivative $p T^{p - 1}$ has no common root with it. The group of its $p$ roots has order $p$ , so any nonidentity root $ζ$ has order $p$ . Thus the chosen-root hypothesis of [F3] holds.

2.1F1F2F3step 1.1

Apply [F2] with $j = 3 = \dim X$ . Using [F1] and taking degree-five cohomology gives an isomorphism from $H_{m o t}^{5, 3} (X, Z / p)$ to $H_{e t}^{5} (X, μ_{p}^{\otimes 3})$ . The inverse of the coefficient isomorphism in [F3] identifies this with $H_{e t}^{5} (X, Z / p)$ . The latter identification depends on the choice of $ζ$ .

3.1F3step 2.1

By affine étale vanishing in [F3], $H_{e t}^{5} (X, μ_{p}^{\otimes 3}) = 0$ because $5 > 3$ . Step 2.1 therefore proves both the displayed isomorphism and the vanishing.

4.1F1F4step 3.1

The first term of the exact sequence in [F4] is zero by [F1] and step 3.1. Exactness therefore makes multiplication by $p$ injective on $H^{6} (C) = H_{m o t}^{6, 3} (X, Z)$ .

5.1F4step 4.1∎

The argument applies to every prime $p$ . The primewise criterion in [F4] now shows that $H_{m o t}^{6, 3} (X, Z)$ is torsion-free. This proves all assertions.

Depends on

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Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 6, proof of Proposition 2.6