Exercise 10, § 5

Denote by k a commutative ring and by E a k-module. For each integer $p \geq 0$ and every k-algebra L we denote by $\gamma_{p,L}$ the mapping of $L \otimes E$ into $L \otimes \Gamma_p(E)$ composed of $\gamma_p : L \otimes E \to \Gamma_p(L \otimes E)$ and the canonical isomorphism of $\Gamma_p(L \otimes E)$ onto $L \otimes \Gamma_p(E)$ (cf. Exercises 2 and 7).

a) Verify that the mappings $\gamma_{p,L}$ form a polynomial law (Exercise 9) of E into $\Gamma_p(E)$, which is homogeneous of degree p ; this law will in what follows be denoted by $\gamma_p$. Show that for every k-module F and every $u \in \mathrm{Hom}(\Gamma_p(E), F)$ the mappings $\varphi_L = (\mathrm{Id}_L \otimes u) \circ \gamma_{p,L}$ form a polynomial law of E into F which is homogeneous of degree p. This law is called the law associated with u.

Let $(x_i)_{i \in I}$ be a family of elements of E and $(a_i)_{i \in I}$ a family with finite support of elements of L. Show that

where $|\xi| = \sum_i \xi(i)$, $a^\xi = \prod_i a_i^{\xi(i)}$ and $\gamma_\xi(x) = \prod_i \gamma_{\xi(i)}(x_i)$.

b) Let Q be a polynomial law of E into a k-module F, homogeneous of degree p. Show that there exists a unique linear mapping $\sigma$ of $\Gamma_p(E)$ into F such that Q is the law associated with a. (Suppose first that E is free on a basis $(x_i)_{i \in I}$; using Exercise 9, show that in this case there exists a unique mapping $\sigma \in \mathrm{Hom}(\Gamma_p(E), F)$ such that Relation (1) of Exercise 9 holds for every algebra L and every family $(a_i)_{i \in I}$ with finite support of elements of L. Treat the general case by introducing a free k-module $E'$ and a surjective homomorphism $h : E' \to E$. The polynomial law $\varphi$, composed with the law defined by h is a polynomial law of $E'$ into F which is associated to a linear mapping $\sigma' : \Gamma_p(E') \to F$; show that $a'$ is the composite of $\Gamma_p(h)$ and a mapping $a : \Gamma_p(E) \to F$ such that Q is associated with $\sigma$ (cf. Exercise 2, f)).