Corollary

Let $A$ be a noetherian local ring, $M$ a finitely generated $A$-module and $J$ an ideal of $A$ generated by a sequence $(x_1, \ldots, x_r)$ of elements of $m_A$ which is both $A$-regular and $M$-regular. Then $\mathrm{dp}_{A/J}(M/JM) = \mathrm{dp}_A(M)$ and $\mathrm{di}_{A/J}(M/JM) = \mathrm{di}_A(M) - r$.

It is enough to treat the case $r = 1$. Put $x = x_1$. The $A$-modules $\mathrm{Ext}^n_{A/xA}(M/xM, \kappa_A)$ and $\mathrm{Ext}^n_A(M, \kappa_A)$ are isomorphic for every integer $n$ (prop. 7); the equality $\mathrm{dp}_{A/xA}(M/xM) = \mathrm{dp}_A(M)$ follows from prop. 4 of No. 3. Analogously, the $A$-modules $\mathrm{Ext}^{n-1}_{A/xA}(\kappa_A, M/xM)$ and $\mathrm{Ext}^n_A(\kappa_A, M)$ are isomorphic for every integer $n$, and the equality $\mathrm{di}_{A/xA}(M/xM) = \mathrm{di}_A(M) - 1$ follows from prop. 6 of No. 3.