Exercise 8, § 7

Let E be a free A-module. (a) Show that in $T^n(E)$, $\mathfrak{g}_n''$ (no. 1) is a free sub-A-module equal to the kernel of the endomorphism $z \mapsto a \cdot z$ and admits a supplement which is a free A-module; $\wedge^n E$ is therefore canonically isomorphic to $A_n''(E)$. If also in A the equation $2\xi = 0$ implies $\xi = 0$, then $A_n''(E) = A_n'(E)$. (b) Suppose that A is a field of characteristic $p > 0$ and that $p \neq 2$. Then the canonical image of $A_p'(E) = A_p''(E)$ in $\wedge^p E$ is zero. (c) Suppose that A is a field of characteristic 2. Then $A_n'(E) = S_n'(E)$, $A_n''(E) = S_n''(E)$, but in general $A_n'(E) \neq A_n''(E)$