By Nicolas Bourbaki
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Extra resources for Algebra II: Chapters 4-7 (Pt.2)
Let (a,),, be a family of elements of K, and suppose that there exists an integer N such that a, = 0 for all n < N. Then the family' (a,Xn), , is summable a,Xn, then u = 0 if and in K ( ( X ) ) (Gen. , 111, p. ) ; put u = , , C nez only if a, = 0 for all n ; otherwise the order of u is the least integer k such that ak# 0. Finally every element of K ( ( X ) ) may be written in a unique fashion in the a,Xn, where the sequence (a,) satisfies a_,= 0 for all sufficiently large form z nsZ n. Since the ring K[X] is a subring of K[[X]], every rational fraction u/v E K ( X ) (u, v being polynomials in X) may be identified with the (generalized) formal power series uv-' of K((X)), which we shall call its expansion at the origin ; the field K ( X ) is thus identified with a subfield of K ( ( X ) ) .
The family (TS"(M)), , is a graduation of type N of the algebra TS(M), and the unit element of T ( M ) is a unit element of TS(M). Let G,, be the set of those (T E G p t q such that , The mapping (u, T ) H U T of G , , x 6,,, into 6,+, is bijective (I, p. 60, q Example 2) ; hence if z E TSP (M) and z' E TS (M), we have PROPOSITION 2. - (i) The A-algebra T S ( M ) is associative, commutative and unital. , pn be integers > 0, and let GP,I ,,, u E 6 ~ , + . . , z, Z1 Z2 . Z n E TS'" ( M ) , then = %PI +..
Let u, v be non-zero elements of A[P]], p = w(u), q = w (v) ; there exists a finite subset J of I such that Let a (resp. b) be the homogeneous component of degree p (resp. q) of cpJ(u) (resp. cpJ(v)). Since J is finite, a and b are polynomials. We have a # 0, b # 0, hence ab # 0 (IV, p. 9, Prop. 8). Hence cp~(u)c p (v) ~ is non-zero, of order p + q. It follows that uv # 0 and w(uv) s p + q ; but clearly w(uv)*p+q. ' 9. The field of fractions of the ring of formal power series in one indeterminate over a field If K is a commutative field, we shall denote by K ( ( X ) ) the field of fractions of the integral domain K[[X]].
Algebra II: Chapters 4-7 (Pt.2) by Nicolas Bourbaki