Résultats de “Pureté” pour les Variétés Lisses sur un Corps Fini
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In this note, we extend the main results of [CT] to more general coefficients than μ n ⨂d . For a constant-twisted sheaf A, with geometric fibre ℤ/ℓ n , coming from the ground field (e.g. A = μ n ⨂i . ), we still prove that, with the notation of [CT], H i (X Zar , H X d+1 (A)) for i = d − 1 and d − 2. (If ℓ = 2, a technical hypothesis on A is necessary; it holds for A = μ n ⨂i .) For an ind-constant-twisted sheaf B, with geometric fibre ℚ ℓ /ℤ ℓ , not isomorphic to ℚ ℓ /ℤ ℓ (d), we prove (under a small technical hypothesis when ℓ = 2) that the sheaf H X d+1 (B) is itself 0, as well as all the terms of its Gersten resolution. The latter result in fact holds for a smooth variety defined over an arbitrary (not necessarily finite) finitely generated field; its proof is much easier than the one for the former result and does not rely on the results of [CT], while the proof of the first result does.
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