Proposition 4

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

(i) one has $\operatorname{dp}_A(M) < n$ ; (ii) one has $\operatorname{Tor}_n^A(M, \kappa_A) = 0$ ; (iii) one has $\operatorname{Ext}_A^n(M, \kappa_A) = 0$ ; (iv) every minimal projective resolution of M has length $< n$.

Assertions (i) $\Rightarrow$ (ii) and (i) $\Rightarrow$ (iii) are immediate (A, X, p. 100, Th. 1). Let L be a minimal projective resolution of M ; if (ii) or (iii) is satisfied, one has $L_n = 0$ by (1). Since every minimal projective resolution of M is isomorphic to L (A, X, p. 54, Prop. 8), one deduces (iv). Implication (iv) $\Rightarrow$ (i) is trivial.