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By Edgar Dehn

Meticulous and whole, this presentation is aimed toward upper-level undergraduate and graduate scholars. It exploresthe simple principles of algebraic conception in addition to Lagrange and Galois thought, concluding with the appliance of Galoisian thought to the answer of specific equations. Many numerical examples, with entire ideas. 1930 variation.

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22 ´ III. INVARIANTS COHOMOLOGIQUES ET PROFONDEUR EXPOSE (iii) La racine de l’annulateur de M, r(Ann M), est l’intersection des id´eaux associ´es ` a M qui sont minimaux (pour la relation d’inclusion dans Ass M). 2. — Soit p un id´eal premier de A, les assertions suivantes sont ´equivalentes : (i) p ∈ Supp M. (ii) Il existe q ∈ Ass M tel que q ⊂ p. (iii) p ⊃ Ann M. (iii bis) p ⊃ r(Ann M). 3. — Soit N un A-module de type fini, on a la formule : Ass HomA (N, M) = Supp N ∩ Ass M. 2. Profondeur Dans tout ce paragraphe, A d´esigne un anneau commutatif, I un id´eal de A, M et N deux A-modules.

An , pour 0 ai < r forment une base de Ir . Ceci dit, si s est un entier, le morphisme de transition tr,r+s : Ir −→ Ir+s (∗) Soient A un anneau, J un id´ eal de A, M un A-module, i ∈ Z ; on posera alors HiJ (M) = HiY (X, F), o` u X = Spec(A), Y = V(J) et F = M. 60 46 ´ IV. ,an +s . Notons que la donn´ee d’un A-homomorphisme w d’un A-module M dans A ´equivaut ` a la donn´ee d’une forme K-lin´eaire w : M → K qui soit continue sur les sousmodules de type fini. Dans le cas M = Hnm (Ωn ), la d´efinition de w ´equivaut donc `a celle d’une forme lin´eaire ρ : Hnm (Ωn ) −→ K, appel´ee forme r´esidu (∗) .

On the structure and ideal theory of complete local rings », Trans. Amer. Math. Soc. 59 (1946), p. 54–106. E. : bien entendu, ce sont les petites limites inductives filtrantes qui sont suppos´ ees exactes ; il faudrait aussi supposer l’existence d’un g´ en´ erateur. Cf. [Tˆ ohoku]. En ce qui concerne la cat´ egorie des modules, suffisante en ce qui nous concerne, on peut aussi se reporter au chapitre 10 de l’Alg` ebre de Bourbaki. 54 42 ´ IV. MODULES ET FONCTEURS DUALISANTS EXPOSE D´emonstration.

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