Remark 5

Let $N$ be a finitely generated $A$-module; the condition $\mathrm{Ext}_{A}^{n}(A/\mathfrak{m}, N) = 0$ for every maximal ideal $\mathfrak{m}$ of $A$ does not necessarily imply $\mathrm{di}_{A}(N) < n$. If, for example, the ring $A$ is local and is not a Gorenstein ring (No. 7, Def. 1), one has $\mathrm{Ext}_{A}^{n}(A/\mathfrak{m}, A) = 0$ for $n < \mathrm{prof}(A)$ but $\mathrm{di}_{A}(A) = +\infty$.