Exercise 12, § 5

Let k be a commutative ring, E a k-module and B a k-algebra (not necessarily associative). Let A be the coproduct of $\Gamma(E)$ (cf. Exercise 3). For any series a, $\sigma' \in S(E, B)$ we denote by $\sigma . \sigma'$ the series m $\circ (a \otimes a') \circ A$, where m denotes the linear mapping of $B \otimes_k B$ into B defined by the product in B. a) Show that the mapping $(\sigma, \sigma') \mapsto \sigma . \sigma'$ defines on $S(E, B)$ a k-algebra structure (cf. Exercise 11). Show that if B is associative (resp. commutative, unital) then $S(E, B)$ is associative (resp. commutative, unital) (cf. III, p. 578-584). b) Assume that B is associative and unital. Show that the invertible elements of $S(E, B)$ are the series $\sigma$ such that $\sigma(1)$ is invertible in B. c) Suppose that E is a free k-module and $(e_i)_{i \in I}$ is a basis of E. For each $a \in \mathbf{N}^{(I)}$ put $e^{(a)} = \Pi \gamma_{a(i)}(e_i)$ and denote by $f^a$ the series over E with values in k defined by $f^a(e^{(\beta)}) = \delta_{a,\beta}$ (Kronecker index) for each $\beta \in \mathbf{N}^{(I)}$ (cf. Exercise 8). Show that $f^a f^\beta = f^{a + \beta}$ for any $a, \beta \in \mathbf{N}^{(I)}$. Deduce that when I is finite, the algebra $S(E, k)$ is isomorphic to the algebra of formal power series $k[[X_i]_{i \in I}]$.