By Arunas Liulevicius (ed.)

ISBN-10: 0821814222

ISBN-13: 9780821814222

ISBN-10: 1119672422

ISBN-13: 9781119672425

ISBN-10: 1277292132

ISBN-13: 9781277292138

ISBN-10: 2019691361

ISBN-13: 9782019691363

ISBN-10: 2542682712

ISBN-13: 9782542682715

ISBN-10: 9219661101

ISBN-13: 9789219661103

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Example text

Il est repr´esent´e par un espace rigide ´etale fini au-dessus de M. Soit U un espace rigide quasicompact. On notera (I, ρ, η) pour une section de MΛ,K sur U . Cela signifie que l’on se donne un mod`ele entier U de U ∼ U rig −−→ U puis une section (I, ρ) ∈ M(U) 32 Chapitre I. D´ecomposition cellulaire de la tour de Lubin-Tate qui d´efinit un ´el´ement s ∈ M(U ) et enfin une section η du produit fibr´e du diagramme suivant GM MΛ,K y s U que l’on notera parfois en ∼ η : Λ −−→ Tp (I) [K] une structure de niveau K sur la fibre g´en´erique de I.

3. 2. Si G est un groupe de Lubin-Tate et K ⊂ Id + π k End(Λ), alors sur XK le groupe G est muni d’une structure de niveau de Drinfeld, de niveau k. C’est une cons´equence du fait que l’espace classifiant des structures de Drinfeld de niveau k est fini au-dessus de X a mˆeme fibre g´en´erique que XK (en fibre g´en´erique toutes les d´efinitions des structures de niveau co¨ıncident) et est donc en-dessous du normalis´e. 3. , la donn´ee pour i entre 1 et n − 1 de nombres rationnels αi , 0 < αi < 1, tels que le polygone commen¸cant en (0, 1), passant par les (q i , αi ) et finissant en (q n , 0) soit convexe.

4). Les valuations des ´el´ements de [π]−1 Fx (ker f \ λ1 λi λ1 1 {0}) sont donc ( n , . . , n ). Or, n < λi , puisque λ1 ≤ , et l’´egalit´e q q q n(q − 1) 24 Chapitre I. D´ecomposition cellulaire de la tour de Lubin-Tate v(xi ) = 1 − i 1 implique que λi ≥ . Donc, i n n(q − q i−1 ) ∀β ∈ ker f \ {0}, ∀α ∈ [π]−1 Fx (ker f \ {0}), v(α) < v(β) On en d´eduit que v(f (α)) = q i v(α), qui sont donc les ( λ1 λi , . . , n−i ), lorsque α q n−i q varie. Or, q i λi+1 ≥ · · · ≥ q i λn > λ1 λi ≥ · · · ≥ n−i q n−i q On en d´eduit aussitˆot que x ∈ ∂n−i D.

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Algebraic Topology by Arunas Liulevicius (ed.)

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