Descente cohomologique by Yves Laszlo

By Yves Laszlo

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Par ailleurs, comme Y et Y co¨ıncident en degr´e ≤ n, le morphisme sqn (φp ) est un isomorphisme. 1), la section diagonale de ϕp = cosq (ϕp ) est un inverse `a homotopie pr`es. 2. 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.

2. 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.

Cosqn+1 (X)p = Xp → cosqn+1 (X) Le morphisme τn+1 s’identifie quant `a lui `a la fl`eche canonique Xn+1 → cosqn (X)n+1 . 2. — Soit n un entier ≥ −1 et τ ∈ HomS (X, X). phisme si p ≤ n et τn+1 est un morphisme de descente cohomologique universelle. Alors, pour tout p, la fl`eche ˜ p cosqn+1 (X)p → cosqn+1 (X) est un morphisme de descente cohomologique universelle. 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 ).

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