Proposition 2

Assume A left noetherian. Let M be a finitely generated a-module and n an integer $\geq0$. The following conditions are equivalent:

(i) $\mathrm{dp}_A(M)\leq n$.

(i bis) M possesses a projective resolution P of length $\leq n$ such that $P_i$ is a finitely generated a-module for each $i$.

(ii) $\operatorname{Ext}_A^r(M,N)=0$ for every finitely generated a-module N and every integer $r>n$.

(iii) $\operatorname{Ext}_A^{n+1}(M,N)=0$ for every finitely generated a-module N.

(iv) $\operatorname{Tor}^A_r(P,M)=0$ for every right a-module P and every $r>n$.

(v) $\operatorname{Tor}^A_{n+1}(A/a,M)=0$ for every finitely generated right ideal $a$ of A.

(i bis) $\Rightarrow$ (i) : this is trivial.

(i) $\Rightarrow$ (ii) : this follows from Lemma 1.

(ii) $\Rightarrow$ (iii) : this is trivial.

(iii) $\Rightarrow$ (i) : by (iii) and X, p. 107, prop. 5, we have $\operatorname{Ext}_A^{n+1}(M,N)=0$ for every a-module N, whence (i) by prop. 1.

(i) $\Rightarrow$ (iv) : this follows from Lemma 1.

(iv) $\Rightarrow$ (v) : this is trivial.

(v) $\Rightarrow$ (i bis) : let (L,d) be a free resolution of M such that $L_r$ is finitely generated for all $r$ (X, p. 53, prop. 6). Set $K=\mathrm{Z}_{n-1}(L)$; then K is finitely generated as a submodule of $L_{n-1}$ and we have an exact sequence

By (v) and X, p. 131, cor. 3, we have $\operatorname{Tor}^A_1(A/a,K)=0$ for every finitely generated right ideal $a$ of A. By th. 2 of X, p. 74, the a-module K is flat; since it is finitely generated, hence of finite presentation (X, p. 10, prop. 5), it is projective (X, p. 13, cor.), so (1) is a projective resolution of M.