Algebre Locale, Multiplicites. Cours au College de France, by Jean-Pierre Serre PDF

By Jean-Pierre Serre

ISBN-10: 3540070281

ISBN-13: 9783540070283

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

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TrK/� (xen )). C’est K−linéaire et ça envoie OK dans n . C’est injectif car TrK/� (xei ) = 0 pour tout i ⇒ Tr(xK) = 0 ⇒ x = 0 ou Tr = 0 absurde ! Donc OK est un −module de type fini. Si I ≤ OK est un idéal, c’est aussi un −module de type fini. iv : soit P un idéal premier non nul. OK /P est entier sur /(P ∩ ). Soit x ∈ P non nul. xn + a1 xn−1 + ... + an = 0 pour certains entiers ai avec an �= 0. Alors 0 �= an ∈ P ∩ . d. premier p. Donc OK /P est un corps et P est maximal. � � � � � � � � Définition 6 Un anneau intègre A qui est intégralement clos, nœthérien et dont tous les idéaux premiers non nuls sont maximaux est un anneau de Dedekind.

D. 2 (Kummer) Soit E/K une extension finie. On suppose que la caractéristique de K est première à n et que K contient une racine primitive n − ième de l’unité. (i) Si E/K est cyclique de degré n, alors il existe α ∈ E tel que E = K(α) et αn ∈ K. 34 (ii) S’il existe α ∈ E tel que E = K(α) et αn ∈ K, alors E/K est galoisienne cyclique et il existe d tel que d|n, E/K est de degré d, αd ∈ K et T d − αd est le polynôme minimal de α sur K. Démonstration : (i) on a N (ζ −1 ) = ζ −n = 1. Donc il existe α ∈ E tel que σα/α = ζ.

Si n < 0, on pose C n (G, A) := 0. Définition 11 Soit n ≥ 0. Soit f ∈ C n (G, A). , gn ) . i=1 On obtient un morphisme dn : C n (G, A) → C n+1 (G, A) pour tout n (si n < 0, dn = 0). 3 Pour tout n, dn dn−1 = 0. ∗. e. i est injective, im i = ker j et j est surjective. , gn+1 )    si i = 0 ; si 1 ≤ i ≤ n ; . , gn+1 ) . i=0 j=0 Or, si j < i, on a : ∂ i ∂ j = ∂ j ∂ i−1 . , gn+1 ) = (−1)i+j ∂ i ∂ j + 0≤i≤n+1 0≤j≤n j

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Algebre Locale, Multiplicites. Cours au College de France, 1957 - 1958 by Jean-Pierre Serre


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