An equivalence for $\operatorname{grade}(I,M)\ge 2$.

Question

Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Show that $\operatorname{grade}(I,M)\ge 2$ iff the canonical homomorphism $M \mapsto\operatorname{Hom}_R(I,M)$ is an isomorphism.

This question is Exercise 1.2.24 in the book: Winfried Bruns and Jürgen Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1998.

Answer 1

This follows from the exact sequence $$0\to \operatorname{Ext}^{0}(R/I, M) \to M\to \operatorname{Hom}_R(I,M)\to \operatorname{Ext}^1(R/I,M)\to 0$$ and Theorem 1.2.5(Rees) (this is usually taken as the definition of grade), p.10 of the same book you mention.