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By D. Mumford

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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

5. — Supposons que pour tout n, les fl`eches Xn+1 → cosqn (X)n+1 soient de S-descente cohomologique universelle. Alors, X → S est de descente cohomologique universelle. DESCENTE COHOMOLOGIQUE 33 D´emonstration. 2, les fl`eches cosqn+1 (X)p → cosqn (X)p sont des ´equivalences de descente cohomologique pour tout p. 1.

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. Le diagramme / ΠX cosqn+1 (X)p  ˜ p cosqn+1 (X) /  ˜ ΠX est cart´esien.

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