Proposition 5

*Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module and $n$ an integer $\geqslant 0$. The following conditions are equivalent :

(i) we have $\mathrm{dp}_A(M) < n$ ; (ii) for every maximal ideal $m$ of $A$, we have $\mathrm{Ext}_A^n(M, A/m) = 0$ (resp. we have $\mathrm{Tor}_n^A(M, A/m) = 0$) ; (iii) for every maximal ideal $m$ of $A$, we have $\mathrm{Ext}_{A_m}^n(M_m, A/m) = 0$ (resp. we have $\mathrm{Tor}_n^{A_m}(M_m, A/m) = 0$).

(i) $\Rightarrow$ (ii) : this is clear. (ii) $\Rightarrow$ (iii) : this follows from Prop. 2 of No. 2. (iii) $\Rightarrow$ (i) : by Prop. 4, condition (iii) implies the inequality $\mathrm{dp}_{A_m}(M_m) < n$ for every maximal ideal $m$ of $A$, and one concludes by Prop. 3.