Exercise 16, § 5

Let $E$ and $F$ be two modules over a commutative ring $k$ and let $p$ be an integer $\geq 0$.

a) Show that if $p \leq 2$, the mapping $\lambda_p$ of $\mathrm{Hom}(\Gamma_p(E), F)$ into $F^E$ which associates with $\sigma$ the mapping $\sigma \circ \gamma_p$ is an injective mapping. b) Show that $A$, has as image $\mathrm{Hom}_k(E, F)$. Show that $A$, has as image the set of all mappings $g$ of $E$ into $F$ such that : (i) $g(ax) = a^2g(x)$ for any $a \in k$ and $x \in E$; (ii) the mapping $(x, y) \mapsto g(x + y) - g(x) - g(y)$ is a bilinear mapping of $E \times E$ into $F$. c) Show that if $E$ is a free $k$-module, then the image of $\lambda_p$ is the module $\mathrm{Pol}_k^p(E, F)$ of homogeneous polynomial mappings of degree $p$ of $E$ into $F$.