New PDF release: Descente cohomologique

Algebraic Geometry

By Yves Laszlo

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5) tr`es commode assurant qu’un S-espace simplicial est de descente cohomologique. 1. — Commen¸cons par un lemme utile. On reprend notre morphisme t : X → Y de S-espaces simpliciaux donnant lieu au diagramme de topos not´e abusivement : X? 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

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

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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