Remark 4

Let $A$ be a noetherian ring and $n$ an integer $\geqslant 0$. If $m$ and $m'$ are two distinct maximal ideals of $A$, the $A$-modules $\mathrm{Ext}_A^n(A/m, A/m')$ and $\mathrm{Tor}_n^A(A/m, A/m')$ are annihilated by $m + m'$, hence are zero. By a proof analogous to that of Corollary 1 of Proposition 4, one deduces from Proposition 5 the equivalence of the following conditions:

(i) one has $\mathrm{dh}(A) < n$ ; (ii) one has $\mathrm{Ext}_A^i(M, N) = 0$ and $\mathrm{Tor}_i^A(M, N) = 0$ for every pair $(M, N)$ of $A$-*modules and every integer* $i \geqslant n$ ; (iii) one has $\mathrm{Tor}_n^A(A/m, A/m) = 0$ for every maximal ideal $m$ of $A$ ; (iv) one has $\mathrm{Ext}_A^n(A/m, A/m) = 0$ for every maximal ideal $m$ of $A$ ; (v) one has $\mathrm{dp}_A(A/m) < n$ for every maximal ideal $m$ of $A$.

In particular, one has $\mathrm{dh}(A) = \sup_m \mathrm{dp}_A(A/m)$, where $m$ ranges over the set of maximal ideals of $A$.