Corollaire

Suppose $A$ is left noetherian and let $n$ be an integer $> 0$. The following conditions are equivalent:

(i) $\mathrm{dh}(A) \leq n$,

(ii) for every pair of a-modules $M$ and $N$ of finite type, we have $\mathrm{Ext}_A^{n+1}(M, N) = 0$,

(iii) for every left a-module $M$ of finite type and every right a-module $P$ of finite type, we have $\mathrm{Tor}_{n+1}^A(P, M) = 0$.

This follows from Props. 2 and 4.