Nicolas Bourbaki's Algebra II: Chapters 4-7 (Pt.2) PDF

By Nicolas Bourbaki

ISBN-10: 3540193758

ISBN-13: 9783540193753

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Extra info for Algebra II: Chapters 4-7 (Pt.2)

Example text

We shall, for any set I, denote by A {I) the set of families ( f )i I satisfying the following conditions : (i) for each i E I, f i is an element of A[[I]] without constant term ; (ii) if I is infinite, f i tends to 0 along the filter of complements of finite subsets of I. The set A {I) is a monoid for the composition law (f, g ) H f o g, with {Xi) I as unit element. The set of invertible elements of A {I) is thus a group. On the other hand, let E be the monoid of all continuous unital endomorphisms of the A-algebra A[[I]] leaving the ideal of all formal power series without constant term invariant.

SYMMETRIC TENSORS AND POLYNOMIAL MAPPINGS Let H be a group and M a left A[H]-module l . We shall denote by M~ the set of all m E M such that hm = m for all h E H ; this is a sub-A-module of M. Let G be a subgroup of H, then MG is a sub-A-module of M containing M ~ . Given m E M', h E H, if x = hG is the left coset of h modulo G, then we have xm = hGm = {hm) . By abuse of notation the element hm of M will be written xm. If h' E H, we have h' (xm ) = (h'x) m . (1) Suppose from now on that G is of finite index in H.

Let (xi)i be a family of elements of E satisfying conditions a) and b) of (i), let IJJ be the homomorphism u I+ ~ ( ( x ~ ) ~of, , A[(Xi), ) ,,I into E, and let V be a neighbourhood of 0 of E which is an ideal of E. By b) there exists a finite subset J of I such that xi E V for all i E I - J. ' E V for all i E J. Let p be the element of N(') such that pi = n - 1 for i E J and pi = 0 for i E I - J. If we define the ideal ap of A [[I]] as at the beginning of No. 2 (IV, p. , NO. 29 This shows that $ is continuous if we equip A [ ( X i ) i ,] , with the topology induced by that of A [ [ I ] ]Since .

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

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