Proposition 4

— Let $M$ be free and $(e_i)_{i \in I}$ a basis of $M$.

(i) For $\nu \in \mathbf{N}^{(I)}$ let $e_\nu = \prod_{i \in I} \gamma_{\nu_i}(e_i)$. Then $(e_\nu)_{\nu \in \mathbf{N}^{(I)}}$ is a basis of the $A$-module $TS(M)$. In particular the algebra $\mathbf{TS}(M)$ is generated by the family of elements $\gamma_k(x)$ for $k \in N$ and $x \in M$.

(ii) For each $p \in N$, $\mathbf{TS}^p(M)$ is a direct factor of the $A$-module $\mathbf{T}^p(M)$.

Let us use the notation of the Remark 2 above. The family $(e_{\rho(1)} \otimes \ldots \otimes e_{\rho(p)})_{\rho \in \mathscr{M}}$ is a basis of $\mathbf{T}^p(M)$. Hence Prop. 4 follows from formula (7) and the following lemma, applied with $H = \mathfrak{S}_p$ and $U = \mathbf{T}^p(M)$.