Read e-book online Algebraic Topology PDF

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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1 Rappel sur le support d’un faisceau cohérent. Soit M un faisceau cohérent sur X. On appelle support de M et on note Supp (M) l’ensemble des points x ∈ X où la fibre Mx est non-nulle. C’est un fermé de X. Lorsque X = Spec(A) et M donné par un module M , le support de M est le fermé V (Ann(M )) défini par l’idéal annulateur de M . – Un morphisme Y −→ X est dit non ramifié au point y ∈ Y si la fibre ΩY /X, y de ΩY /X en y est nulle. e. si ΩY /X = 0. 3 Remarques. (i) On voit que le lieu de non-ramification d’un morphisme est un ouvert.

Plus précisément, on a 32 Université Pierre et Marie Curie Master de Mathématiques Proposition. – Avec les notations ci-dessus, le morphisme XI −→ Pn × X est une immersion fermée, et identifie XI à l’adhérence schématique du graphe de U −→ Pn . Démonstration. Le morphisme en question est une immersion fermée puisqu’il est induit par OX [T0 , · · · , Tn ] −→ ReesI (OX ) qui est par construction surjectif. La restriction de ce morphisme à U est le graphe de U −→ Pn . Or U est schématiquement dense dans XI , étant le complémentaire d’un diviseur.

L’anneau local A est régulier). 39 Université Pierre et Marie Curie Master de Mathématiques Démonstration. Soit d = dimA/m (m/m2 ), et f1 , · · · , fd des éléments de m formant une base de m/m2 sur A/m. Par le lemme de Nakayama, on voit qu’ils engendrent m. 3) A/m[T1 , · · · , Td ] −→ grm (A). Par la théorie du polynôme de Hilbert-Samuel (cf Eisenbud, ch. 12), on sait qu’il existe un polynôme P de degré dim(A) − 1 tel que pour n entier assez grand, on ait dim(mn /mn+1 ) = P (n). Supposons m régulier.

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Algebraic Topology by Arunas Liulevicius (ed.)


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