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External affine étale vanishing at invertible primes

PDF p. 5: Proposition 2.6: torsion freeness

Statement

If X is affine of finite type and dimension d over an algebraically closed field, ℓ is invertible on X, and j≥0, then Heˊti(X,μℓ⊗j)=0(i>d). No smoothness or reducedness hypothesis is required.

External input

This is the precise Artin affine étale vanishing input cited as SGA4 XIV Corollary3.2 and used in the published report Theorem4 proof and its affine application. It is imported, not proved here. The twist is an étale torsion local system, so is among the coefficients of Artin's theorem.

Depends on

Used by · 3

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Artin affine étale vanishing, as explicitly stated and applied in Elmanto’s published Zero cycles report · Theorem4 proof, printed1995/PDF47, and affine application printed1996/PDF48; cited primary source SGA4 tome3, Exposé XIV Corollaire3.2

H^i_et(X,µ_ℓ^⊗j)=0 for i>dim(X).

Hypotheses: Affine finite-type X over an algebraically closed field, ℓ invertible, torsion étale coefficients of order prime to characteristic (here µ_ℓ^⊗j).

Reading limits: Read the exact statement/application in the published report; original SGA4 proof not read or reconstructed. The substantial Artin result is explicitly imported through this authoritative primary application. A Milne EC1.pdf download attempt returned404 and supplies no reading credit.

J.S. Milne, Lectures on Étale Cohomology, version2.21 (22 March2013) · Theorem15.1, p.105; nilpotent invariance of the étale category, p.22

For a variety over an algebraically closed field, cd(X)<=2dim(X), and if X is affine cd(X)<=dim(X); nilpotent thickenings do not change the etale category.

Hypotheses: Algebraically closed base field, affine finite-type variety; nilpotent thickening comparison extends to nonreduced finite-type schemes; only invertible-prime torsion coefficients used here.

Reading limits: Read full Theorem15.1 statement and opening discussion, and the full nilpotent-invariance statement p22. Did not reconstruct its field/cohomological-dimension proof or claim original SGA4 proof reading.

Sources