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Affine étale vanishing and the actual localized evaluation

PDF p. 6: Affine Artin vanishing

Statement

Assume the Axiom of Choice and the fixed universe and site-size bounds of the supplied derived-localization constructions. Let $k$ be algebraically closed, $X / k$ affine of finite type with $\dim X \leq 3$ , and $p$ an invertible prime. Then $H_{e t}^{a} (X, Z / p) = 0$ for $a \geq 4$ . Let $F$ be the prime subfield of $k$ . Use the site of all qcqs $F$ -schemes in the fixed universe, with topology generated by Nisnevich covers and abstract blowup families whose closed immersion and proper morphism are finitely presented. Write $L_{c d h}^{F} = L_{c d h}^{o, F}$ for ordinary $D (Z)$ -valued cdh sheafification. The natural map induced by truncation is an equivalence: $(L_{c d h}^{F} τ^{\leq 3} R Γ_{e t} (-, Z / p)) (X) \tilde{\to} R Γ_{e t} (X, Z / p) .$ For this affine $X$ , its left side has $H^{5} = 0$ . The evaluation equivalence itself holds for every finite-type $X / k$ of dimension at most three, including nonaffine $X$ . Thus the full characteristic-zero affine-threefold application in Proposition 2.6 is retained. Localization is on the entire qcqs prime-field site; the equivalence is asserted at these evaluation objects, without asserting a global equivalence on that site.

Statement references

The source qcqs cdh site

Cancellation at the required dimension-bounded evaluation objects

Affine Artin bound and its consequence after localization

Facts & Assumptions

[A1]

The Axiom of Choice and the fixed universe and site-size bounds are inherited by the finite-coefficient locality, slice localization and derived-evaluation constructions. The Axiom of Choice

Given: The field, prime, scheme and size conventions of the statement; affineness is needed for the cohomological vanishing, while the evaluation comparison only needs finite type and dimension at most three.

Goal: Establish the actual evaluation equivalence and the affine $H^{a}$ and localized $H^{5}$ vanishings.

[F1]

The source qcqs cdh site fixes the entire qcqs prime-field cdh site and its finitely presented abstract blowup generators.

[F2]

Slice localization and the actual root-of-unity coefficient map compares constant and weight-three Tate coefficients on the slice over $X$ , commutes restriction with ordinary localization including its units, and identifies the actual induced maps at $X$ .

[F3]

An independent full equivalence into the truncated cdh localization supplies the full canonical equivalence $γ_{X} : Z (3)_{m o t} (X) / p ≃ (L_{c d h}^{F} τ^{\leq 3} B_{3}) (X)$ , where $B_{3} = R Γ_{e t} (-, μ_{p}^{\otimes 3})$ . Independently, The full top-weight map with its exact factorization supplies the full canonical equivalence $β_{X} : Z (3)_{m o t} (X) / p ≃ B_{3} (X)$ and the identity $β_{X} = ϵ_{X}^{(3)} γ_{X}$ for the actual map $ϵ_{X}^{(3)}$ induced by truncation. Both equivalences apply to finite-type $X / k$ with $\dim X \leq 3$ without affineness, smoothness or reducedness.

[F4]

Cancellation at the required dimension-bounded evaluation objects packages cancellation and the coefficient change as the exact constant-coefficient evaluation equivalence for ordinary source sheafification, with the same Choice and size hypotheses.

[F5]

Affine Artin bound and its consequence after localization supplies the affine Artin bound $H_{e t}^{a} (X, Z / p) = 0$ for $a > \dim X$ , including singular and nonreduced affine finite-type schemes.

Proof

1.1givenA1F1F3

Work first with any finite-type $X / k$ of dimension at most three, viewed as a qcqs $F$ -scheme on the site in [F1]. By [F3], $β_{X} = ϵ_{X}^{(3)} γ_{X}$ , and both $β_{X}$ and $γ_{X}$ are full equivalences. Cancellation gives $ϵ_{X}^{(3)} = β_{X} γ_{X}^{- 1}$ as an equivalence. The argument uses the actual source maps and their factorization.

2.1A1F2F4step 1.1choose

Choose a primitive $p$ th root in the algebraically closed field $k$ . By [F2], the slice coefficient equivalence conjugates the twisted evaluation map in step 1.1 to the constant-coefficient map displayed in the statement, compatibly with ordinary localization units. Equivalently this is the precise cancellation result [F4]. It holds for nonaffine as well as affine $X$ in this dimension range. The root is used only over $X$ ; no root in $F$ or change to a dimension-bounded localization site is required.

3.1givenA1F5step 2.1∎

If $X$ is affine, [F5] gives $H_{e t}^{a} (X, Z / p) = 0$ for $a > \dim X$ , hence for every $a \geq 4$ . Taking $H^{5}$ in the equivalence of step 2.1 gives zero on its left side since $5 > \dim X$ . This includes the characteristic-zero affine threefold used in the original Proposition 2.6.

The evaluation argument uses the separate canonical comparison in Theorem 4 of the published Oberwolfach report, at an invertible prime, through the existing top-weight supplier. The report's characteristic-prime square and its Remark 5 duality display are not premises here. The exact factorization is checked against Elmanto-Morrow v1, especially the proof of Theorem 4.33(1).

The original article's passage from affine Artin vanishing to a global truncation equivalence on the entire qcqs site remains a recorded proof-step gap. The present proof instead establishes the needed evaluation comparison before using the affine vanishing bound. The substantial published comparisons and Artin theorem remain external inputs with the reading limits recorded above; this draft supplies no new proof or acceptance of those theorems.

Source and scope notes

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Sources

Fasel–Yakerson · p. 6, proof of Proposition 2.6 (arXiv:2610.22679v1)
Elden Elmanto, Zero cycles on singular varieties, Oberwolfach Report 36/2025 · Theorems 3(1) and 4, printed pp. 1994-1995 (PDF pp. 46-47)
Elmanto-Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Section 2.1, p. 11; section 3.3 and Theorem 3.5(3), pp. 14-15; Theorems 4.10(4), 4.24(3), 4.33(1), pp. 21, 28, 31
J. S. Milne, Lectures on Etale Cohomology, version 2.21, 22 March 2013 · Theorem 15.1, p. 105; nilpotent invariance, p. 22