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Algebra II: Chapters 4-7 (Pt.2) by Nicolas Bourbaki

By Nicolas Bourbaki

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Additional resources for Algebra II: Chapters 4-7 (Pt.2)

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The family (e,(,) @ ... @ e,@)),,A is a basis of T P ( M ) . Hence Prop. 4 follows from formula ( 7 ) and the following lemma, applied with H = G p and U = T P ( M ) . Lemma 1. - Let H be a finite group and U a left A [HI-module. Suppose that the A-module U has a basis B which is stable under the operations o f H in U , and put Cl = B/H. For each w E R let u , = C b ; then hew is a basis o f the A-module u H . (i) (u,),, (ii) For each o E R let V , be a point o f o ; put w' = w - {v,) and B' = U w', then B' is a basis o f a supplementary subspace for U" in U.

Iv) For each basis (ei)i of N such that I , of M there exists a family ( u ,), ), I,I ,of elements = for all (Ai ) E A('). ( i ) + (ii) : let g satisfy ( i ) , then there exists a linear mapping g ' o f T 4 ( M ) into N such that g(x,, x,, .. , x , ) = g l ( x , O x 2 @... , x, E M. , X ) = g l ( x 0x 0 ... 0 x ) = g ' ( y , ( x ) ) ; q and on writing h = g ' IT S ( M ) we see that condition (ii) holds. (ii) + (i) and (iv) : let h satisfy the conditions o f (ii). By Prop. 4, (ii) ( I V , p. 47) there exists a linear mapping g ' o f T q ( M ) into N such that h = g ' ITSq(M).

Homogeneous polynomial mappings P RO POS ITIO N 13. - Let M and N be A-modules, q an integer z 0 , and f a mapping of M into N . , x ) for all x E M. q (ii) There exists a linear mapping h of T S ( M ) into N such that f ( x ) = h ( y , ( x ) ) for all x E M. (iii) There exists a basis (e,)i I I = o f elements of M and a family (u,),, of N such that , for all ( A i ) E A('). (iv) For each basis (ei)i of N such that I , of M there exists a family ( u ,), ), I,I ,of elements = for all (Ai ) E A(').

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