Proposition 1

Let $M$ be an $A$-module and $n$ an integer $\geq 0$. The following conditions are equivalent : (i) $\mathrm{dp}_A(M) \leq n$ (i.e. (def. 1), $M$ has a projective resolution of length $\leq n$); (ii) $\mathrm{Ext}_A^r(M, N) = 0$ for every $A$-module $N$ and every integer $r > n$; (iii) $\mathrm{Ext}_A^{n+1}(M, N) = 0$ for every $A$-module $N$; (iv) for every exact sequence

où les $P_i$ sont projectifs, $K$ is projective.

(i) ⇒ (ii): this follows from Lemma 1. (ii) ⇒ (iii): this is trivial. (iii) ⇒ (iv): in the situation of (iv), for every $A$-module $N$ there is an isomorphism of $\mathrm{Ext}_A^1(K, N)$ onto $\mathrm{Ext}_A^{n+1}(M, N)$ (X, p. 128, Corollary 4); if (iii) holds, then

for every $N$ and $K$ is projective (X, p. 93, Proposition 10). (iv) ⇒ (i): consider the exact sequence (X, p. 50) If (iv) holds, then $Z_{n-1}(M)$ is projective and $M$ has a projective resolution of length $\leq n$.