Descente cohomologique by Yves Laszlo PDF

By Yves Laszlo

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D´emonstration. — On peut supposer p > n + 1. On ´ecrit alors cosqn+1 (X)p = lim Xq ←− [q]→[p] q≤n+1 comme le noyau de la double fl`eche ΠX = d´ ef Xq ⇒ Xi = ΞX α [q]→[p] q≤n+1 [i] →[j] [p] j≤n+1 o` u la composante αX d’indice α ∈ Hom[p] ([i], [j]) de la double fl`eche est la double fl`eche form´ee d’une part du morphisme ΠX → Xi de projection d’indice [i] → [p] et, d’autre part, du morphisme ΠX → Xj → Xi , compos´e de la projection d’indice [j] → [p] et de α ∈ Hom(Xj , Xi ). 3. — Soit p un entier naturel.

T ??  f / Y . 1. — Supposons que tp : Xp → Yp soit un isomorphisme pour p ≤ n. Alors, pour tout Faisceau F de SF , et at induit un isomorphisme ∼ τ

D´emonstration. 1). Ici, on s’int´eresse avant tout `a t = f• : X• → S∆ . On va approximer t de proche en proche : on a une factorisation X · · · → cosqn+1 (X) → cosqn (X) · · · → cosq−1 (X) = S∆ . c) qu’on a une identification cosqm ◦ cosqn = cosqn pour m ≥ n. La fl`eche cosqn+1 (X) → cosqn (X) s’interpr`ete alors comme une fl`eche ˜ τ : cosqn+1 (X) → cosqn+1 (X) ˜ = cosqn (X). Remarquons que τp est un isomorphisme si p ≤ n, cette fl`eche s’identifiant o` u l’on a pos´e X a l’identit´e de ` ˜ p=X ˜ p = cosqn (X)p = Xp .

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Descente cohomologique by Yves Laszlo

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