Exercise 12, § 4

Let $A$ be an integrally closed Noetherian domain and $M, N$ two torsion-free finitely generated $A$-modules of the same rank $r$ such that $N \subset M$; let $j : N \to M$ be the canonical injection. (a) Show that, for all $k$, $\bigwedge^k j : \bigwedge^k N \to \bigwedge^k M$ is pseudo-injective, that $\operatorname{Coker}\left( \bigwedge^k j \right)$ is a torsion $A$-module and that:

(b) Using (a), show that, if $M$ is a torsion-free finitely generated $A$-module, the torsion submodule of $\bigwedge^k M$ is pseudo-zero for all $k$ and where $M$ is of rank $r$; in particular, $c\left( \bigwedge^r M \right) = c(M)$.