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By Yves Laszlo

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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 .

4. — Soit f : Y• → Y• un S-morphismes d’espaces simpliciaux. On suppose que les fl`eches Y• → cosqn+1 (Y• ), Y• → cosqn+1 (Y• ) sont des isomorphismes et fp de descente cohomologique pour tout p ≥ 0. Alors, si Y• est de S-descente cohomologique, il en est de mˆeme de Y• . D´emonstration. — On regarde le diagramme cart´esien (changement de base par π) / Y • Y•• = cosq0 (Y• ×Y• Y• /Y• ) f˜   f / Y• Y•• = cosq0 (Y• /Y• ) qu’on factorise en Y•• f˜  Y•• π ˜ π / Y •∆ / Y•∆ Y Y / Y . •  f / Y• Il suffit de se convaincre que f˜, π ˜ , π sont des ´equivalences de S-descente cohomologique.

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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Categories: Algebraic Geometry