Exercise 42, § 1

Commutative Algebra, IX, p. 56

Let $A$ be a ring and $T$ an indeterminate. Denote by $\Lambda(A)$ (or $\Lambda_T(A)$ ) the set $1 + TA[[T]]$ of formal series with coefficients in $A$ whose constant term is equal to 1; this is a subgroup of the multiplicative group of $A[[T]]$ . Define

L : \Lambda(A) \to A^J , \quad L(f) = (L_n(f))_{n \in J} ,

- T \frac{df}{dT}/f = \sum_{n \in J} L_n(f)(-T)^n .

a) Now prove that one has $L(fg) = L(f) + L(g)$ for any $f, g \in \Lambda(A)$ .

b) If $A$ is a $\mathbf{Q}$ -algebra, then $L$ is bijective, its inverse $\varepsilon$ being given by

\varepsilon(a) = \exp(- \sum_{n \in J} \frac{a_n}{n} (-T)^n)

for

a \in A^J

Solution

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