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Affine Artin bound and its consequence after localization

PDF p. 6: Proposition 2.6 proof

Statement

Assume the Axiom of Choice and the fixed universe and site-size bounds of the supplied derived-localization constructions. Let $k$ be an algebraically closed field, $p$ a prime invertible in $k$ , and $X$ an affine finite-type $k$ -scheme. Then $H_{e t}^{a} (X, Z / p) = 0$ for $a > \dim X$ , hence for $a \geq 4$ when $\dim X \leq 3$ . If in addition $\dim X \leq 3$ , let $F$ be the prime field of $k$ , put $B = R Γ_{e t} (-, Z / p)$ on the supplied qcqs prime-field cdh site, and let $L_{c d h}^{o, F}$ denote ordinary source sheafification. Then $(L_{c d h}^{o, F} τ^{\leq 3} B) (X)$ has $H^{5} = 0$ . Both assertions include singular and nonreduced schemes.

Facts & Assumptions

[A1]

Choice and the stated bounds are inherited through the finite-coefficient locality, ordinary slice localization and derived-evaluation suppliers used below. The Axiom of Choice

Given: The algebraically closed field $k$ , the invertible prime $p$ , the affine finite-type $k$ -scheme $X$ , and the Choice, universe and site-size conventions in the statement. For the localized-complex conclusion also assume $\dim X \leq 3$ .

Goal: Prove the affine cohomological bound and the dimension-bounded localized $H^{5}$ vanishing.

[F1]

External affine étale vanishing at invertible primes gives the affine Artin bound for $μ_{p}^{\otimes 3}$ under exactly the stated field, prime and finite-type hypotheses, without smoothness or reducedness.

[F2]

Slice localization and the actual root-of-unity coefficient map gives the primitive-root coefficient equivalence on the slice over $X$ and its compatibility with ordinary localization units.

[F3]

Cancellation at the required dimension-bounded evaluation objects identifies $(L_{c d h}^{o, F} τ^{\leq 3} B) (X)$ with $B (X)$ when $X$ is finite type over $k$ and $\dim X \leq 3$ .

Proof

1.1givenF1F2A1choose

Choose a primitive $p$ th root in $k$ . The coefficient equivalence of [F2], evaluated at $X$ , identifies $R Γ_{e t} (X, Z / p)$ with $R Γ_{e t} (X, μ_{p}^{\otimes 3})$ . Apply [F1] with twist $3$ : its hypotheses hold for the stated affine finite-type $X / k$ and invertible $p$ . Taking cohomology yields $H_{e t}^{a} (X, Z / p) = 0$ for $a > \dim X$ , in particular for $a \geq 4$ if $\dim X \leq 3$ .

2.1givenF3A1step 1.1∎

Under the additional hypothesis $\dim X \leq 3$ , apply the actual evaluation equivalence [F3] for the ordinary source reflector. It gives $H^{5} ((L_{c d h}^{o, F} τ^{\leq 3} B) (X)) ≅ H_{e t}^{5} (X, Z / p) = 0$ by step 1.1, since $5 > \dim X$ . This proves the localized consequence at $X$ ; it does not assert a global equivalence between $B$ and its unlocalized truncation on the unbounded qcqs site.

Depends on

Used by · 1

Affine étale vanishing and the actual localized evaluation

Dependency tree · 3 levels

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