Corollary

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module and $q$ an ideal of $A$ such that $M/qM$ is of finite length. Then $d_q(M)$ is the least upper bound of the dimensions $\dim_{A_m}(M_m)$, where $m$ ranges over the finite set $S = \operatorname{Supp}(M) \cap V(q)$, and $e_q(M)$ is the sum of the $e_{q_m}(M_m)$ extended to those elements $m$ of $S$ for which one has $\dim_{A_m}(M_m) = d_q(M)$.

For each integer $n$, the length of $M/q^nM$ is the sum of the $\operatorname{long}_{A_m}(M_m/q^n_m M_m)$ (IV, $§ 2$, No. 5, cor. 1 to prop. 7 and corollary to prop. 8). Consequently, one has $H_{M,q} = \sum_{m \in S} H_{M_m,q_m}$, whence the corollary.