Algebre commutative : Chapitres 1 a 4 by Nicolas Bourbaki

By Nicolas Bourbaki

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1, 5 2 PROPOSITION 10. - Soient A, B deux anneaux, E un A-module à droite, F un B-module à droite, G un (B, A)-bimodule. Supposons que F soit plat. Alors, s i E est de type fini (resp. de présentation finie) l'homomorphisme canonique (8) est injectif (resp. bijectif). Considérons A, B, F, G comme fixés, et, pour tout A-module à droite E, posons T(E) = F c h Hom, (E, G), T1(E) = Hom, (E, F 6% G ) et notons v, I'homomorphisme (8) ; pour tout homomorphisme v : E 4 E' de A-modules à droite, posons T(v) = 1, €3 Hom (v, 1,) et T1(v) = Hom (v, IF@ 1,).

On dit qu'un A-module à droite E est fidèlement plat s'il vérifie les quatre propriétés équivalentes de la prop. 1. On définit de même les A-modules à gauche fidèlement plats ; il est clair que pour qu'un A-module à gauche E soit fidèlement plat, il faut et il sufit que E , considéré comme Ao-module à droite, soit fidèlement plat. Remarque. -- Si E est un A-module fidèlement plat, E est un A-module fidèle : en effet, si un élément a E A est tel que z a = O pour tout x E E, l'homothétie h : b + ba dans A est telle que lx@ h = O ; d'où h = O par la propriété c) de la prop.

Si on prend M = E, on en déduit (puisque i @ 1, = 1, @3j) que j est injectif (no 1, prop. 2). L'assertion (ii) résulte de (i) en prenant F = &/a, e t (iii) résulte de (ii) en prenant a = 10 Enfin, si m est un idéal à gauche maximal de A, on a 1. -1 p(Bm) = m en vertu de (ii), e t par suite Bm # B. , chap. 1, $ 8, no 7, th. 2) ; on a m c;(n) e t comme p ( l )e n , 2 n'appartient pas à-;(n). P a r suite$@) = m. Lorsque A et B vérifient les conditions de la prop. 8, on identifie d'ordinaire A à un sous-anneau de B a u moyen de p.

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