Exercise 15, § 5

With the notation as in Ex. 14, let $E, F, G$ be three k-modules, $\sigma \in S(E, F)$, $\tau \in S(F, G)$, and suppose that $\sigma$ is of order $\geq 1$. The series $\tau \circ A(\sigma) \in S(E, G)$ is called the composition of $a$ and $\tau$; it is denoted by $\tau \circ a$. a) Let $\varphi$ be a polynomial law of $E$ into $F$ (Ex. 9) and $\psi$ a polynomial law of $F$ into $G$, and denote by $a_\varphi$ and $\sigma_\psi$ the series associated with $\varphi$ and $\psi$ respectively. Show that if the homogeneous component of degree 0 of $\varphi$ is zero, then $\sigma_\varphi$ is of order $\geq 1$ and $\sigma_\psi \circ \sigma_\varphi$ is the series associated with the polynomial law $\psi \circ \varphi$. b) Show that if $a$ is of order $\geq p$ and $\tau$ is of order $\geq q$, then $\tau \circ \sigma$ is of order $\geq pq$. c) Show that for every integer $p$, $(\tau \circ \sigma)^{(p)} = \tau^{(p)} \circ A(\sigma)$. Deduce that if $\tau$ is of order $\geq 1$ and if $H$ is a k-module, then for any $\rho \in S(G, H)$ we have $(\rho \circ \tau) \circ a = \rho \circ (\tau \circ \sigma)$. d) Suppose that $G$ is a k-algebra. Show that the mapping $\tau \mapsto \tau \circ A(\sigma)$ is a homomorphism of the algebra $S(F, G)$ into the algebra $S(E, G)$.