Exercise 8, § 5

Let k be a commutative ring and E a k-module. a) Show that there exists a unique homomorphism g of the algebra $\Gamma(E)$ (Exercise 2) into the algebra $TS(E)$ of symmetric tensors over E (IV, p. 43) such that

for any $x \in E, p \in \mathbf{N}$. Show that if E is a free k-module with basis $(e_i)_{i \in I}$, then g is an isomorphism of graded bigebras (cf. Exercise 3 and IV, p. 51) and the elements $\gamma_a = \prod \gamma_{a(i)}(e_i)$, where $a \in \mathbf{N}^{(I)}$, form a basis of $\Gamma(E)$.

b) Let $\xi$ be a mapping of $(1, n)$ into $\mathbf{N}$ and let $p = \sum_{i=1}^n \xi(i)$. Show that for every sequence $x_1, x_2, \ldots, x_n$ of elements of E, the image of $\gamma_{\xi(1)}(x_1) \ldots \gamma_{\xi(n)}(x_n)$ under g is $\sum \varphi_{\varphi(1)} \otimes \ldots \otimes \varphi_{\varphi(p)}$ where $\varphi$ runs over the set of all mappings of $(1, p)$ into $(1, n)$ such that Card $\varphi^{-1}(i) = \xi(i)$ for each $i \in (1, n)$ (cf. IV, p. 45, Prop. 3). c) Let f be the mapping of $\otimes^p E$ into $\Gamma_p(E)$ defined by $f(x_1 \otimes \ldots \otimes x_p) = \gamma_1(x_1) \ldots \gamma_1(x_p)$ for any $x_1, \ldots, x_p \in E$. Show that for each tensor $t \in \otimes^p E$, $(g \circ f)(t)$ is the symmetrization of t. Show that for each $u \in \Gamma_p(E)$ we have $(f \circ g)(u) = p! \ u$.