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Characteristic-prime degree-five vanishing for singular affine threefolds

PDF p. 5: Proposition 2.6: torsion freeness

Statement

Let $X = Spec A$ be an affine finite-type threefold over an algebraically closed field of characteristic $p > 0$ . Then $H_{m o t}^{5, 3} (X, Z / p) = 0$ .

Facts & Assumptions

[F1]

External canonical top-weight motivic-to-syntomic comparison supplies the canonical EM motivic-to-syntomic equivalence in weight $3 \geq \dim X$ .

[F2]

Affine characteristic-p syntomic amplitude without regularity supplies the upper bound four for affine weight-three mod- $p$ syntomic cohomology.

Proof

Given: $X$ as stated, including singular and nonreduced $X$ .

Goal: Prove the exact mod- $p$ vanishing needed by Bockstein.

1.1F1

View $A$ as an $F_{p}$ -algebra. Evaluate [F1] on $X$ : $Z (3)_{m o t} (X) \otimes^{L} Z / p ≃ F_{p} (3)_{s y n} (A)$ . This is the comparison of the specified complexes, not an extrapolation of a degree- $\leq 3$ truncated comparison.

2.1F2step 1.1∎

Take degree-five cohomology and apply [F2]. It is zero. No K-theoretic regularity or quasisyntomic assumption enters.

Depends on

Used by · 1

Torsion freeness of top threefold motivic cohomology

Dependency tree · 3 levels

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