Corollary 1

Let A be a noetherian local ring and n an integer $\geq 0$. The following conditions are equivalent :

(i) one has $\operatorname{dh}(A) < n$ ; (ii) one has $\operatorname{Ext}_A^i(M, N) = 0$ and $\operatorname{Tor}_i^A(M, N) = 0$ for every pair $(M, N)$ of A-modules and every integer $i \geq n$ ; (iii) one has $\operatorname{Tor}_n^A(\kappa_A, \kappa_A) = 0$ ; (iv) one has $\operatorname{Ext}_A^n(\kappa_A, \kappa_A) = 0$ ; (v) one has $\operatorname{dp}_A(\kappa_A) < n$.

It is clear that (i) implies (ii) and that (ii) implies (iii) and (iv). By Prop. 4 applied to the A-module $\kappa_A$, each of conditions (iii) and (iv) implies (v). Let us prove that (v) implies (i) : if $\operatorname{dp}_A(\kappa_A) < n$, one has $\operatorname{Tor}_n^A(M, \kappa_A) = 0$ for every A-module M ; hence every finitely generated A-module is of projective dimension $< n$ (Prop. 4), which implies $\operatorname{dh}(A) < n$ (A, X, p. 138, Prop. 4).