Exercise 1, § 5

(Algebra of sequences of exponential type) Given two integers $p, q \in N$, we denote by $((p, q))$ the binomial coefficient $\frac{(p+q)!}{p! q!}$. Let k be a commutative ring and B an associative commutative unital k-algebra. By a sequence of exponential type of elements of B we understand a sequence $a = (a_p)_{p \in \mathbf{N}}$ such that $a_1 = 1$, and $a_p a_q = ((p, q)) a_{p+q}$ for any $p, q \in N$. The set of all sequences of exponential type of B is written $\mathscr{E}(B)$. a) Verify that there is on $\mathscr{E}(B)$ a unique k-algebra structure such that

for any sequence $a, b \in \mathscr{E}(B)$ and $p \in \mathbf{N}$, $\lambda \in k$. The mapping $a \mapsto a$, is a homomorphism of the algebra $\mathscr{E}(B)$ into $B$. b) Suppose that $k$ contains a subring isomorphic to $\mathbf{Q}$. Show that for every $x \in B$ the sequence $f(x)$ defined by $f(x)_p = \frac{1}{p!} x^p$ is of exponential type. Show that $f$ is an isomorphism of $B$ onto $\mathscr{E}(B)$. c) Let $B'$ be an associative commutative unital $k$-algebra and let $h$ be a homomorphism of $B$ into $B'$. For each $a \in \mathscr{E}(B)$ the sequence $a'$ defined by $a'_p = h(a_p)$ for any $p \in \mathbf{N}$ belongs to $\mathscr{E}(B')$. Show that the mapping $\mathscr{E}(h): a \mapsto a'$ is a homomorphism of $\mathscr{E}(B)$ into $\mathscr{E}(B')$. Verify that $\mathscr{E}(g \circ h) = \mathscr{E}(g) \circ \mathscr{E}(h)$ when $g$ and $h$ are composable $k$-algebra homomorphisms. d) Let $E$ be a $k$-module and $TS(E)$ the algebra of symmetric tensors on $E$. Show that for each $x \in E$, $(\gamma_p(x))$ is a sequence of exponential type in the algebra $TS(E)$ (IV, p. 45).