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

By Nicolas Bourbaki

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3) Let (Mi)i be a finite family of free A-modules and u : Mi -+ N a ,, n j s J multilinear mapping ; then u is polynomial of degree Card(J). 56 05 POLYNOMIALS AND RATIONAL FRACTIONS 4) Let (Xi)i,, be a family of indeterminates, N an A-module and u E N [(Xi)i ,] a homogeneous polynomial of degree q. The mapping (xi)i I H u ( ( x ~ ) ,) ~ ,of A(') into N is a homogeneous polynomial mapping of degree q : this is seen at once by condition (iii) of Prop. 13. If I is finite, every homogeneous polynomial mapping of degree q of A(') = A' into N is of that form.

Xu ; In other words, the constant term of AUuis the coefficient of Xu in u. Since the mapping u H u(X + Y ) of A[[X]] into A[[X, Y]] is continuous, the mappings u H AVu of A[[X]] into itself are again continuous. As in the case of polynomials (IV, p. 7) we can prove the formulae The binomial formula (I, p. 99, Cor. 2) gives the following value for AUuwhen Consider in particular the case v = ei, that is vi = 1, vj = 0 for j # i . We shall U differently, Diu is the coefficient of Yi in u (X + Y ).

This property is independent of the basis chosen for M and it justifies the terminology (< polynomial mapping P. PROPOSITION 17. - Let M be a pee A-module and B an associative, c o m m u t d v e and unital A-algebra. Then PolA(M, B ) is a sub-B-algebra o f the algebra Map (M, B 1. This follows from Def. 4 and Prop. 13, (iv) ( I V , p. 54). PROPOSITION 18. - Let M , N, P be A-modules, and assume that M and N are free. I f f E Pol(M, N ) , g E Pol(N, P), then g o f E Pol(M, P ) . , y ) for all y E N.

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