By Jean-François Dat
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Extra info for Géométrie Algébrique 2 [Lecture notes]
Sample text
Soit A un anneau et I un idéal. On peut munir tout A-module M de la filtration I-adique (décroissante) (I n M )n∈N M ⊃ IM ⊃ I 2 M ⊃ · · · . Lorsque A = M , c’est une filtration de l’anneau A. Rappelons deux propriétés de ces filtrations : i) Lemme d’Artin-Rees : Supposons A noethérien et M de type fini. Soit N un sousmodule de M . Alors il existe un entier n0 tel que pour tout n n0 on a I n M ∩ N = I n−n0 (I n0 M ∩ N ). ii) Théorème d’intersection de Krull : Supposons A noethérien, M de type fini, et I contenu dans tous les idéaux maximaux de A.
Iii) ⇔ i). Supposons iii) ; comme dans l’exemple plus haut, on voit que OY,y est plate et non ramifiée sur OX,x .
Ces morphismes se recollent en U := i Xsi −→ Pn . On ne peut pas prolonger ce morphisme à X, mais on peut le prolonger après éclatement. En effet, soit F ⊂ L le sous-OX -module engendré par les si . Il n’est plus inversible, mais π il le devient sur l’éclatement X −→ X de l’idéal F ⊗ L⊗−1 ⊂ OX . Les sections π ∗ si de π ∗ L engendrent donc un sous-module inversible et définissent un morphisme X −→ Pn . ∼ Par construction π induit un isomorphisme π −1 (U ) −→ U , et identifiant U à π −1 (U ), le morphisme ainsi construit prolonge le précédent.