Proposition 5

— Let $M$ be a free $A$-module, $k$ an integer $\geq 0$, $P$ the sub-$A$-module of $\mathbf{TS}^k(M)$ generated by $\gamma_k(M)$. Assume $A$ to be an infinite integral domain. Then for each $z \in \mathbf{TS}^k(M)$ there exists $a \in A - \{0\}$ such that $az \in P$.

Let $K$ be the field of fractions of $A$. We identify $\mathbf{TS}^k(M)$ with a sub-A-module of the vector K-space $V = \mathbf{TS}^k(M) \otimes_A K$ (Prop. 4, and II, p. 314). We have to show that this vector K-space is generated by $\gamma_k(M)$, that is, every K-linear form $f$ on V satisfying $f(\gamma_k(M)) = 0$ is zero. Let $(e_i)_{i \in I}$ be a basis of M, and define the e, as in Prop. 4. For any $(\alpha_i) \in A^{(I)}$ we have, on taking (6) into account,

By Cor. 2 of IV, p. 18 it follows that $f(e_\nu) = 0$ for all $\nu \in N^{(I)}$, whence $f = 0$.