Exercise 17, § 1

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module. Now prove that $M$ is reflexive if and only if the following two conditions are satisfied:

(i) for every $p \in \mathrm{Spec}(A)$ such that $\mathrm{prof}(A_p) \leqslant 1$, the $A_p$-module $M_p$ is reflexive ; (ii) for every $p \in \mathrm{Spec}(A)$ such that $\mathrm{prof}(A_p) \geqslant 2$, one has $\mathrm{prof}_{A_p}(M_p) \geqslant 2$.

(Under the hypotheses (i) and (ii), proving that the kernel, then the cokernel of the canonical homomorphism $M \to M^{**}$ have no associated prime ideals.)