Exercise 11, § 5

We keep the notation of Exercise 10. A linear mapping of $\Gamma(E)$ into F will be called-a series over E with values in F. The series over E with values in F form a k-module denoted by $S(E, F)$. For $n \in \mathbf{N}$ we say that the series $\sigma \in S(E, F)$ is of order $\geq n$ if $\mathrm{Ker}(\sigma) \supset \Gamma_p(E)$ for all $p < n$. In what follows we shall equip $S(E, F)$ with the topological group structure having as neighbourhood base of 0 the k-modules consisting of the series of order $\geq n$ ($n \in \mathbf{N}$).

Let Q be a polynomial law of E into F. Show that there exists a unique series $\sigma \in S(E, F)$ such that: (i) for every k-algebra L and every $u \in L \otimes E$ the family $(\mathrm{Id}_L \otimes \sigma) \gamma_{p,L}(u)$ ($p \in \mathbf{N}$) has finite support ; (ii) $\varphi_L(u) = \sum_p (\mathrm{Id}_L \otimes \sigma) \circ \gamma_{p,L}(u)$. (Apply the result of Exercise 10, b)). This series $\sigma$ is said to be associated to the law Q.