By Nicolas Bourbaki
Read or Download Algebra II: Chapters 4-7 (Pt.2) PDF
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Extra resources for Algebra II: Chapters 4-7 (Pt.2)
For every integer n a 0 let b, be the set of formal power series u E A [[I]] such that w (u) 3 n. The sequence (b,),, is a fundamental , No. 2 FORMAL POWER SERIES system of neighbourhoods of 0 in A[[I]]. Therefore a family of elements uA of A[[I]] (A E L ) is summable if and only if for every n E N the set of A E L such that w ( u , ) < n is finite. , , be two PROPOSITION 1. - Let (u,),, and (v,),, elements of A[[I]]. , , summable families of is summable and we have Let (resp. (p,,,),, ,(I)) be the family of coefficients of u , (resp.
Arguing as before, we can show that A (h) belongs to b, - for every homogeneous polynomial h of degree n 3 1. Now let u E A [[XI] and let u, be the homogeneous component of degree n of u. Since A (u,) E b, - for n z=1, the family (A (u,)), , is summable in A[[X]] and we can define a derivation D of A[[X]] into itself by , , We have D (b,) c b, - hence D is a continuous endomorphism of the additive group of A[[X]]. The mapping a : ( u , v ) ~ D ( u v ) - u D ( v ) - D ( u ) v of A[[X]] x A [[XI] into A[[X]] is continuous and zero on A[X] x A [XI.
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('). ( 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,, ..