Analyse Différentielle by Valentin Poenaru (auth.)

By Valentin Poenaru (auth.)

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0 ° Cette topologie est s6par6e si et seulement si : < = pl {oi i=I Si Krull". A est local, la topologie '~A-adique" s'appelle "topologie de 46 Si M est un A-module prenant comme syst&me fondamental on d@finit la topologie de voisinage On peut montrer que, si est s6par@ pour la topologie M est de 0 ( M de Krull les A-fini et A ~k(A) sur M en M o est noeth@rien, M de Krull. : lim A / ~ k ( A ) K-- s'appelle le sgpar@-compl6t6 de A . On a une application A la compl@tion. , Xn) et ~ = l'application (formelle) qui consiste ~ associer ~ f sa s6rie de Taylor ~ l'origine.

K r4sidu du pOle (zj + P Z Z j£A' k=1 n'a pas on a P(x)dx = i X (la somme des r@sidus de a(X,u) = i [ Z P(Z) I ~. aZk-2ki J + I Zk-Ej-2ki = ("Rzj-RZk) + iiImz. + Imz. - 2K) J J I i(Imz. + [mz k- 2k) J 6 @ + (~zk- ~zj) $"~I~zj. s. Done : (Imz + Imz k - 2k) J . . 1~j,k~p oh Izj - ~k #jk = I <---j ((j,k) E A) ~jk = 0 ~ ~jk = -I ~ Done : q(k,u) _ ((j £ A , k £ A') ou (j £ A' , k £ A)) . ((j,k) E A') est une somme de ~jk p2 quantit6s ( I ~ 3 -x) + (Im~k - x ) (R~j - R~) 2 + ((Im~j - ~) + (Imzk - ~))2 >,o est : 31 Chaque quantit@ est Mais il y e n I ~< a tune : (j,k = Jo oh : I Ims.

K r4sidu du pOle (zj + P Z Z j£A' k=1 n'a pas on a P(x)dx = i X (la somme des r@sidus de a(X,u) = i [ Z P(Z) I ~. aZk-2ki J + I Zk-Ej-2ki = ("Rzj-RZk) + iiImz. + Imz. - 2K) J J I i(Imz. + [mz k- 2k) J 6 @ + (~zk- ~zj) $"~I~zj. s. Done : (Imz + Imz k - 2k) J . . 1~j,k~p oh Izj - ~k #jk = I <---j ((j,k) E A) ~jk = 0 ~ ~jk = -I ~ Done : q(k,u) _ ((j £ A , k £ A') ou (j £ A' , k £ A)) . ((j,k) E A') est une somme de ~jk p2 quantit6s ( I ~ 3 -x) + (Im~k - x ) (R~j - R~) 2 + ((Im~j - ~) + (Imzk - ~))2 >,o est : 31 Chaque quantit@ est Mais il y e n I ~< a tune : (j,k = Jo oh : I Ims.

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