Proposition 9

Tag 06CD, permanent. Cite it as [Bourbaki, Tag 06CD].

Algebra, chapter IV, § 4. Formal power series

— Let $(u_\lambda)_{\lambda \in L}$ be a summable family of elements of $A[[I]]$ without constant term and $D$ a continuous derivation of the $A$ -algebra $A[[I]]$ . If $f = \prod_{\lambda \in L} (1 + u_\lambda)$ (IV, p. 27, Prop. 2), then the family $(Du_\lambda/(1 + u_\lambda))_{\lambda \in L}$ is summable and we have

D(f)/f = \sum_{\lambda \in L} D(u_\lambda)/(1 + u_\lambda) .

If $g$ and $h$ are two invertible elements of $A[[I]]$ , then

D(gh) = h \cdot Dg + g \cdot Dh

whence on division by $gh$ ,

D(gh)/gh = D(g)/g + D(h)/h .

For every finite subset $M$ of $L$ put $f_M = \prod_{\lambda \in M} (1 + u_\lambda)$ . From (21) we deduce by induction on Card $M$ the relation

D(f_M)/f_M = \sum_{\lambda \in M} D(u_\lambda)/(1 + u_\lambda)

This proves Prop. 9 when L is finite. Now suppose that L is infinite and write $\mathfrak{F}$ for the filtered ordered set of finite subsets of L. We have $\lim_{\mathfrak{F}} f_M = f$ , and hence (IV, p. 30 remark)

D(f)/f = \lim_{\mathfrak{F}} D(f_M)/f_M .

Now Prop. 9 follows by passage to the limit in (22).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

IV, p. 27, Prop. 2