Exercise 14, § 11

Let M be a free A-module of dimension $n > 1$. Show that every isomorphism $\psi$ of $\bigwedge^p(M)$ onto $\bigwedge^{n-p}(M^*)$ such that $\bigwedge^{n-p}(\tilde{u}) \circ \psi = \psi \circ \bigwedge^p(u)$ for every automorphism $u$ of M, of determinant equal to 1, is one of the isomorphisms $\phi_p$ defined in Proposition 12 of no. 11 (proceed as in § 7, Exercise 4).