Corollary

Suppose A semi-local and let q be an ideal of definition of A. a) One has $e_q(A) = \sum_p e_q(A/p)$, where p runs through the set of minimal prime ideals of A such that $\dim(A/p) = \dim(A)$. b) Suppose A integral and let M be a finitely generated A-module such that $\dim_A(M) = \dim(A)$. Then one has $e_q(M) = \operatorname{rg}(M).e_q(A)$.