Proposition 16

— *Let M be a free A-module, N an A-module, q a positive integer and u the canonical homomorphism of Hom ($\mathbf{TS}^q(M)$, N) into $\mathbf{Pol}^Y(M, N)$.

(i) If A is an infinite integral domain and N is torsion-free, then u is an isomorphism.

(ii) If the mapping $y \mapsto q!y$ in N is injective, u is an isomorphism.

In the two cases of the proposition we have to prove that u is injective, that is, every linear mapping f of $\mathbf{TS}^q(M)$ into N which is zero on $\gamma_q(M)$, vanishes.

Assume A to be an infinite integral domain and N torsion-free. For every $z \in \mathbf{TS}^q(M)$ there exists $a \in A - \{0\}$ such that $az$ is an A-linear combination of elements of y, (M) (IV, p. 47, Prop. 5). Hence $af(z) = f(az) = 0$, and so $f(z) = 0$.

Suppose next that the mapping $y \mapsto q!y$ in N is injective, then by IV, p. 45, Prop. 3, (v), f vanishes on s . T^q(M). Hence if $z \in \mathbf{TS}^q(M)$, we have $q!f(z) = f(sz) = 0$, and so $f(z) = 0$.