Exercise 4, § 4

Let $A$ be a noetherian ring of dimension $\leqslant 2$. Show that in order that $A$ be regular, it is necessary and sufficient that the dual of every finitely generated $A$-module be projective (let $\mathfrak{p}$ be a prime ideal of $A$; if the second condition is satisfied, first prove that $A_{\mathfrak{p}}$ is regular when $\operatorname{prof}(A_{\mathfrak{p}}) \leqslant 1$, then that $\mathrm{dp}_{A_{\mathfrak{p}}}(\kappa(\mathfrak{p})) = 2$ when $\operatorname{prof}(A_{\mathfrak{p}}) = 2$).