# Algèbre locale, multiplicités by Jean-Pierre Serre, Pierre Gabriel

By Jean-Pierre Serre, Pierre Gabriel

This variation reproduces the 2d corrected printing of the 3rd version of the now vintage notes through Professor Serre, lengthy demonstrated as one of many general introductory texts on neighborhood algebra. Referring for historical past notions to Bourbaki's "Commutative Algebra" (English variation Springer-Verlag 1988), the publication focusses at the quite a few measurement theories and theorems on mulitplicities of intersections with the Cartan-Eilenberg functor Tor because the important suggestion. the most effects are the decomposition theorems, theorems of Cohen-Seidenberg, the normalisation of jewelry of polynomials, size (in the feel of Krull) and attribute polynomials (in the feel of Hilbert-Samuel).

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Additional info for Algèbre locale, multiplicités

Example text

Mn+ k et la surjectivit~ tel que de la suite de Cauchy ce qui mcntre bien que logie de de ; on prend alors limites darts on a Mn n . Nn+k+ I y ~ N l'une des Mn est ferm@, . Done U(Mn)=N n , strict surjectif. La topo- , et c'est donc un module complet. Corollaire I: Soient filtre s@par~, (ni) (xi)i ~ I un anneau filtr@ complet, , les xi dans xi M u__n_nA-module une famille finie d'@l~ments de une famille fini~ d'entiers tels ~ue l'ima~e de gr(M) A grn(~ ) l engendrent . Si les M , et ~ ~i M , e_t_t 9 On note i l engendrent le gr(A)-module xi~ Mn est complet.

I' ~2'''''a-~ on a sont les annulateurs de a = ~I ~] ~ 2 ~ . . ,M s et on obtient ainsi la comme intersection d'id@aux gtrangers deux & deux et qui n ' o n t p l n s eux-mGmes de telle reprgsentation. Nous allons monter au ciel en nous tirant par nos lacets de chaussures (la d@monstration est celle du th4or&me chinois; on peut l'ignorer sans pr@judice): Supposons d'abord donn@ une d@composition les M. a__s = ~Ti = ~ 1 ~ ' " e M i - 1 , ... F'~ (O:Ms) = (O:M) = (O:M1)/~} Alors: aI j~i --3a @ Mi+1""(B Ms , p o ~ I~ , et i < s .

1 a+bs = I sont confondus N i = ~i/~ , isomor~hes comme A-module, i ~ j , A/~ 9 Alors & , . On en MT. d. 1 Corollaire A/~ s m/s T V(Mi) AT -module l j ~ i suffit de montrer que cette application Soit donc dgsignons par sous ~acent au aij+aji = I, aij Alors ci-dessus, de la rgunion des idgaux premiers de est isomorphe D'abord . A/~ i . des est m~me 1-30 ll en r4sulte en particulier de A/~ 3) Modules de lon~ueur finie. que le treillis des id4aux est compos~ direct des treillis des id4aux des Nous supposerons A/~ i connus les r~sultats de Bourbaki, Alg~bre, chapitre I.